Computational Photonics, An Introduction With MATLAB

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http://www.cambridge.org/9781107005525

Computational Photonics

A comprehensive manual on the efficient modelling and analysis of photonic devicesthrough building numerical codes, this book provides graduate students and researcherswith the theoretical background and MATLAB programs necessary for them to start theirown numerical experiments.

Beginning by summarizing topics in optics and electromagnetism, the book discussesoptical planar waveguides, linear optical fibres, the propagation of linear pulses, laserdiodes, optical amplifiers, optical receivers, the finite-difference time-domain method, thebeam propagation method and some wavelength division devices, solitons, solar cells andmetamaterials.

Assuming only a basic knowledge of physics and numerical methods, the book is ideal forengineers, physicists and practicing scientists. It concentrates on the operating principlesof optical devices, as well as the models and numerical methods used to describe them.

Marek S. Wartak is a Professor in the Department of Physics and Computer Science, WilfridLaurier University, Waterloo, Ontario. He has over 25 years of experience in semiconductorphysics, photonics and optoelectronics, analytical methods, modelling and computer-aideddesign tools.

Computational PhotonicsAn Introduction with MATLAB

MAREK S. WARTAKWilfrid Laurier University

C A M B R I D G E U N I V E R S I T Y P R E S S

Cambridge, New York, Melbourne, Madrid, Cape Town,Singapore, Sao Paulo, Delhi, Mexico City

Cambridge University PressThe Edinburgh Building, Cambridge CB2 8RU, UK

Published in the United States of America by Cambridge University Press, New York

www.cambridge.orgInformation on this title: www.cambridge.org/9781107005525

C© M. S. Wartak 2013

This publication is in copyright. Subject to statutory exceptionand to the provisions of relevant collective licensing agreements,no reproduction of any part may take place without the written

permission of Cambridge University Press.

First published 2013

Printed and bound in the United Kingdom by the MPG Books Group

A catalogue record for this publication is available from the British Library

Library of Congress Cataloguing in Publication dataWartak, Marek S., author.

Computational photonics : an introduction with MATLAB / Marek S. Wartak,Department of Physics and Computer Science, Wilfrid Laurier University,

Waterloo, Ontario, Canada.pages cm

Includes bibliographical references and index.ISBN 978-1-107-00552-5 (hardback)

1. Optoelectronic devices – Mathematical models. 2. Photonics – Mathematics.3. MATLAB. I. Title.TK8304.W37 2012

621.3815′2 – dc23 2012037576

ISBN 978-1-107-00552-5 Hardback

Cambridge University Press has no responsibility for the persistence oraccuracy of URLs for external or third-party internet websites referred to

in this publication, and does not guarantee that any content on suchwebsites is, or will remain, accurate or appropriate.

http://www.cambridge.org

http://www.cambridge.org/9781107005525

Contents

Preface page xi

1 Introduction 11.1 What is photonics? 11.2 What is computational photonics? 21.3 Optical fibre communication 51.4 Biological and medical photonics 121.5 Photonic sensors 121.6 Silicon photonics 131.7 Photonic quantum information science 14References 14

2 Basic facts about optics 172.1 Geometrical optics 172.2 Wave optics 212.3 Problems 30Appendix 2A: MATLAB listings 31References 34

3 Basic facts from electromagnetism 353.1 Maxwell’s equations 353.2 Boundary conditions 363.3 Wave equation 383.4 Time-harmonic fields 393.5 Polarized waves 423.6 Fresnel coefficients and phases 443.7 Polarization by reflection from dielectric surfaces 483.8 Antireflection coating 503.9 Bragg mirrors 533.10 Goos-Hanchen shift 583.11 Poynting theorem 593.12 Problems 603.13 Project 60Appendix 3A: MATLAB listings 61References 63

v

vi Contents

4 Slab waveguides 644.1 Ray optics of the slab waveguide 644.2 Fundamentals of EM theory of dielectric waveguides 694.3 Wave equation for a planar wide waveguide 714.4 Three-layer symmetrical guiding structure (TE modes) 724.5 Modes of the arbitrary three-layer asymmetric planar waveguide in 1D 754.6 Multilayer slab waveguides: 1D approach 794.7 Examples: 1D approach 854.8 Two-dimensional (2D) structures 884.9 Problems 924.10 Projects 92Appendix 4A: MATLAB listings 93References 104

5 Linear optical fibre and signal degradation 1065.1 Geometrical-optics description 1065.2 Fibre modes in cylindrical coordinates 1105.3 Dispersion 1235.4 Pulse dispersion during propagation 1275.5 Problems 1285.6 Projects 129Appendix 5A: Some properties of Bessel functions 129Appendix 5B: Characteristic determinant 130Appendix 5C: MATLAB listings 131References 137

6 Propagation of linear pulses 1386.1 Basic pulses 1386.2 Modulation of a semiconductor laser 1436.3 Simple derivation of the pulse propagation equation in the presence of

dispersion 1466.4 Mathematical theory of linear pulses 1486.5 Propagation of pulses 1526.6 Problems 155Appendix 6A: MATLAB listings 156References 165

7 Optical sources 1677.1 Overview of lasers 1677.2 Semiconductor lasers 1727.3 Rate equations 1817.4 Analysis based on rate equations 1877.5 Problems 1967.6 Project 196

vii Contents

Appendix 7A: MATLAB listings 196References 202

8 Optical amplifiers and EDFA 2048.1 General properties 2058.2 Erbium-doped fibre amplifiers (EDFA) 2098.3 Gain characteristics of erbium-doped fibre amplifiers 2138.4 Problems 2158.5 Projects 215Appendix 8A: MATLAB listings 215References 222

9 Semiconductor optical amplifiers (SOA) 2239.1 General discussion 2239.2 SOA rate equations for pulse propagation 2289.3 Design of SOA 2319.4 Some applications of SOA 2339.5 Problem 2369.6 Project 236Appendix 9A: MATLAB listings 236References 238

10 Optical receivers 24010.1 Main characteristics 24110.2 Photodetectors 24210.3 Receiver analysis 25110.4 Modelling of a photoelectric receiver 25710.5 Problems 25810.6 Projects 258Appendix 10A: MATLAB listings 258References 260

11 Finite difference time domain (FDTD) formulation 26211.1 General formulation 26211.2 One-dimensional Yee implementation without dispersion 26611.3 Boundary conditions in 1D 27211.4 Two-dimensional Yee implementation without dispersion 27511.5 Absorbing boundary conditions (ABC) in 2D 27711.6 Dispersion 28011.7 Problems 28011.8 Projects 281Appendix 11A: MATLAB listings 281References 286

viii Contents

12 Beam propagation method (BPM) 28812.1 Paraxial formulation 28812.2 General theory 29212.3 The 1 + 1 dimensional FD-BPM formulation 29912.4 Concluding remarks 30612.5 Problems 30712.6 Project 308Appendix 12A: Details of derivation of the FD-BPM equation 308Appendix 12B: MATLAB listings 310References 314

13 Somewavelength division multiplexing (WDM) devices 31613.1 Basics of WDM systems 31613.2 Basic WDM technologies 31713.3 Applications of BPM to photonic devices 32313.4 Projects 325Appendix 13A: MATLAB listings 325References 329

14 Optical link 33114.1 Optical communication system 33114.2 Design of optical link 33314.3 Measures of link performance 33614.4 Optical fibre as a linear system 33814.5 Model of optical link based on filter functions 34014.6 Problems 34414.7 Projects 344Appendix 14A: MATLAB listings 345References 348

15 Optical solitons 35115.1 Nonlinear optical susceptibility 35115.2 Main nonlinear effects 35215.3 Derivation of the nonlinear Schrodinger equation 35315.4 Split-step Fourier method 35715.5 Numerical results 36115.6 A few comments about soliton-based communications 36415.7 Problems 364Appendix 15A: MATLAB listings 365References 366

16 Solar cells 36816.1 Introduction 36816.2 Principles of photovoltaics 370

ix Contents

16.3 Equivalent circuit of solar cells 37316.4 Multijunctions 376Appendix 16A: MATLAB listings 379References 381

17 Metamaterials 38417.1 Introduction 38417.2 Veselago approach 38817.3 How to create metamaterial? 38917.4 Some applications of metamaterials 39517.5 Metamaterials with an active element 40017.6 Annotated bibliography 401Appendix 17A: MATLAB listings 401References 403

Appendix A Basic MATLAB 406A.1 Working session with m-files 407A.2 Basic rules 409A.3 Some rules about good programming in MATLAB 410A.4 Basic graphics 412A.5 Basic input-output 416A.6 Numerical differentiation 417A.7 Review questions 418References 418

Appendix B Summary of basic numerical methods 420B.1 One-variable Newton’s method 420B.2 Muller’s method 422B.3 Numerical differentiation 425B.4 Runge-Kutta (RK) methods 432B.5 Solving differential equations 433B.6 Numerical integration 435B.7 Symbolic integration in MATLAB 438B.8 Fourier series 438B.9 Fourier transform 441B.10 FFT in MATLAB 443B.11 Problems 446References 446

Index 448

Preface

Photonics (also known as optoelectronics) is the technology of creation, transmission,detection, control and applications of light. It has many applications in various areas ofscience and engineering fields. Fibre optic communication is an important part of photonics.It uses light particles (photons) to carry information over optical fibre.

In the last 20 years we have witnessed the significant (and increasing) presence of photon-ics in our everyday life. The creation of the Internet and World Wide Web was possible dueto tremendous technical progress created by photonics, development of photonic devices,improvement of optical fibre, wavelength division multiplexing (WDM) techniques, etc.The phenomenal growth of the Internet owes a lot to the field of photonics and photonicdevices in particular.

This book serves as an attempt to introduce graduate students and senior undergraduatesto the issues of computational photonics. The main motivation for developing an approachdescribed in the present book was to establish the foundations needed to understand prin-ciples and devices behind photonics.

In this book we advocate a simulation-type approach to teach fundamentals of photonics.We provide a self-contained development which includes theoretical foundations and alsothe MATLAB code aimed at detailed simulations of real-life devices.

We emphasize the following characteristics of our very practical book:

• learning through computer simulations• writing and analysing computer code always gives good sense of the values of all param-

eters• our aim was to provide complete theoretical background with only basic knowledge

assumed• the book is self-contained in a sense that it starts from a very basic knowledge and ends

with the discussion of several hot topics.

The author believes that one can learn a lot by studying not only a theory but also byperforming numerical simulations (numerical experiments) using commercial software ordeveloped in-house. Here, we adopt a view that the best way will be to develop a softwareourselves (as opposed to commercial). Also there is one more component in the learningprocess, namely real-life experimentation. That part is outside the scope of the presentbook.

The goal was to equip students with solid foundations and underlying principles byproviding a theoretical background and also by applying that theory to create simpleprograms in MATLAB which illustrate theoretical concepts and which allow students toconduct their own numerical experiments in order to get better understanding.

xi

xii Preface

Alternatively, students can also use octave (www.octave.org) to run and to conductnumerical experiments using programs developed and distributed with this book. At theend, they can try to develop large computer programs by combining learned ideas. At thisstage, I would like to remind the reader that according to Donald Knuth (cited in ComputerAlgebra Handbook, J. Grabmeier, E. Kaltofen and V. Weispfenning, eds., Springer, 2003)the tasks of increasing difficulty are:

i) publishing a paperii) publishing a book

iii) writing a large computer program.

The book has evolved from the lecture notes delivered in the Department of ElectricalEngineering, University of Oulu, Finland, Institute of Physics, Wroclaw University ofTechnology and Department of Physics and Computer Science, Wilfrid Laurier University,Waterloo, Canada.

The book has been designed to serve a 12-week-long term with several extra chapterswhich allow the instructor more flexibility in choosing the material. An important part ofthe book is the MATLAB code provided to solve problems. Students can modify it andexperiment with different parameters to see for themselves the role played by differentfactors. This is a very valuable element of the learning process.

The book also contains a small number of solved problems (labelled as Examples) whichare intended to serve as a practical illustration of discussed topics. At the end of eachchapter there is a number of problems intended to test the student’s understanding of thematerial.

The book may serve as a reference for engineers, physicists, practising scientists andother professionals. It concentrates on operating principles of optical devices, as well as theirmodels and numerical methods used for description. The covered material includes fibre,planar waveguides, laser diodes, detectors, optical amplifiers and semiconductor opticalamplifiers, receivers, beam propagation method, some wavelength division devices, finite-difference time-domain method, linear and nonlinear pulses, solar cells and metamaterials.

The book can also be used as a textbook as it contains questions, problems (both solvedand unsolved), working examples and projects. It can be used as a teaching aid at both under-graduate and graduate levels as a one-semester course in physics and electrical engineeringdepartments.

The book contains 17 chapters plus two appendices. The first two chapters summarizebasic topics in optics and electromagnetism. The book also contains a significant numberof software exercises written in MATLAB, designed to enhance and help understanding thediscussed topics. Many proposed projects have been designed with a view to implementthem in MATLAB.

Some numerical methods as well as important practical devices were not considered.Those developments will be included in the second edition. The author would like to wisha potential reader the similar joy of reading the book and experimenting with programs ashe had with writing it.

Finally, I welcome all type of comments from readers, especially concerning errors,inaccuracies and omissions. Please send your comments to [email protected].

xiii Preface

Requirements I do not assume any particular knowledge since I tried to cover all basictopics. Some previous exposure to optics and electromagnetic theory will be helpful.

Acknowledgements Thanks are due to Professors Harri Kopola and Risto Myllyla(University of Oulu) and Professor Jan Misiewicz (Wroclaw University of Technology) forarranging delivery of lectures. I wish also to thank my former students in Finland, Polandand Canada for comments on various parts of lecture notes. I am grateful to my wife forreading the whole manuscript and Dr Jacek Miloszewski for preparing the index and figurefor the cover.

My special thank you goes to people at Cambridge, especially Dr Simon Capelin andMrs Antoaneta Ouzounova for their patience and good advice, and Lynette James for herexcellent work.

The book support grant from the Research Office of Wilfrid Laurier University is ac-knowledged. Support for my research from various Canadian organizations including fed-eral NSERC as well as from province of Ontario is gratefully acknowledged.

I dedicate this book to my family.

1 Introduction

In this introductory chapter we will try to define computational photonics and to position itwithin a broad field of photonics. We will briefly summarize several subfields of photonics(with the main emphasis on optical fibre communication) to indicate potential possibilitieswhere computational photonics can significantly contribute by reducing cost of designingnew devices and speeding up their development.

1.1 What is photonics?

We start our discussion from a broader perspective by articulating what photonics is, whatthe current activities are and where one can get the most recent information.

Photonics is the field which involves electromagnetic energy, such as light, where thefundamental object is a photon. In some sense, photonics is parallel to electronics whichinvolves electrons. Photonics is often referred to as optoelectronics, or as electro-optics toindicate that both fields have a lot in common. In fact, there is a lot of interplay of photonicsand electronics. For example, a laser is driven by electricity to produce light or to modulatethat light to transmit data.

Photonics applications use the photon in a similar way to that which electronic applica-tions use the electron. However, there are several advantages of optical transmission of dataover electrical. Furthermore, photons do not interact between themselves (which is bothgood and bad), so electromagnetic beams can pass through each other without interactingand/or causing interference.

Even with the telecommunication ‘bubble’ which happened some ten years ago, thesubfield of photonics, namely optical fibre communication, is still a very important segmentof photonics activities. As an example, a single optical fibre has the capacity to carry about3 million telephone calls simultaneously. But, due to a crisis around 2000, many newapplications of photonics have emerged or got noticed. Bio-photonics or medical photonicsare amongst the most important ones.

Before we try to define computational photonics, let us concentrate for a moment onwhat photonics is at the time of writing (Winter, 2011). In the following paragraphs, wecite some information from the relevant conferences.

Some of the well-established conferences which summarize research and applicationsin a more traditional approach to photonics, and also dictate new directions are: PhotonicsWest, held every January in California (here is the 2011 information [1]); Photonics East,

1

2 Introduction

taking place in the eastern part of the USA in the Fall; Optical Fibre Conference (OFC)taking place in March every year [2].

A new conference, the Photonics Global Conference (PGC), is a biennial event heldsince 2008 [3]. The aim of this conference is to foster interactions among broad disciplinesin photonics with concentration on emerging directions. Symposia and Special Sessionsfor 2011 are summarized below:

• Symposia: 1. Optofluidics and Biophotonics. 2. Fibre-Based Devices and Applications.3. Green Photonics. 4. High-Power Lasers and Their Industrial Applications. 5. Metama-terials and Plasmonics. 6. Nanophotonics. 7. Optical Communications and Networks.

• Special Sessions: 1. Quantum Communications. 2. Photonics Crystal Fibres and TheirApplications. 3. Photonics Applications of Carbon Nanotube and Graphene. 4. TerahertzTechnology. 5. Diffuse Optical Imaging.

By looking at the discussed topics, one can get some sense about current activities inphotonics.

1.2 What is computational photonics?

Computational photonics is a branch of physics which uses numerical methods to studyproperties and propagation of light in waveguiding structures. (Here, light is used in a broadsense as a replacement for electromagnetic waves.) Within this field, an important part isplayed by studies of the behaviour of light and light-matter interactions using analyticaland computational models. This emerging field of computational science is playing acritically important role in designing new generations of integrated optics modules, longhaul transmission and telecommunication systems.

Generally, computational photonics is understood as the ‘replacement’ of the experimen-tal method, where one is performing all the relevant ‘experiments’ on computer. Obviously,such an approach reduces development cost and speeds-up development of new products.

We will attempt to cover some of those developments. The field is, however, so broadthat it is impossible to review all of its activities. Naturally, selection of the topics reflectsauthor’s expertise.

As a separate topic in photonics, one selects integrated photonics where the concentrationis on waveguides, simulations of waveguide modes and photonic structures [4]. The centralrole is played there by a beam-propagation method which will be discussed later in thebook.

1.2.1 Methods of computational photonics. Computational electromagnetics

According to Joannopoulos et al. [5], in a broad sense there are three categories of problemsin computational photonics: frequency-domain eigensolvers, frequency-domain solversand time-domain simulations. Those problems are extensively discussed in many sourcesincluding [5], also [6] and [7].

3 What is computational photonics?

A recent general article by Gallagher [8] gives an introduction to the main algorithmsused in photonic computer aided design (CAD) modelling along with discussion of theirstrengths and weaknesses. The main algorithms are:

• BPM – Beam Propagation Method• EME – Eigenmode Expansion Methods• FDTD – Finite Difference Time Domain.

The above methods are compared against speed, memory usage, numerical aperture, therefractive index contrasts in the device, polarization, losses, reflections and nonlinearity.The author’s conclusion is that none of the discussed algorithms are universally perfect forall applications.

We finish this section by indicating that application of computational electromagneticsgoes well beyond photonics, see the discussion by Jin [6]. In fact it has an extremely widerange of applications which goes from the analysis of electromagnetic wave scatteringby various objects, antenna analysis and design, modelling and simulation of microwavedevices, to numerical analysis of electromagnetic interference and electromagnetic com-patibility.

1.2.2 Computational nano-photonics

This subfield has emerged only in recent years. As the dimensions of photonics devicesshrink, it plays an increasingly important role. Nanostructures are generally consideredas having sizes in the order of the wavelength of light. At those wavelengths, multiplescattering and near-field effects have a profound influence on the propagation of light andlight-matter interaction. In turn, this leads to novel regimes for basic research as well asnovel applications in various disciplines.

For instance, the modified dispersion relation of photonic crystals and photonic crystalfibres leads to novel nonlinear wave propagation effects such as giant soliton shifts and su-percontinuum generation with applications in telecommunication, metrology and medicaldiagnostics. Owing to the complex nature of wave interference and interaction processes,experimental studies rely heavily on theoretical guidance both for the design of such sys-tems as well as for the interpretation of measurements. In almost all cases, a quantitativetheoretical description of such systems has to be based on advanced computational tech-niques that solve the corresponding numerically very large linear, nonlinear or coupledpartial differential equations.

University research on photonic crystal structures applied to novel components in opticalcommunication, boomed since it has become possible to create structures on the nanometrescale. Nanotechnology offers a potential for more efficient integrated optical circuitry atlower cost – from passive elements like filters and equalizers to active functions such asoptical switching, interconnects and even novel lasers. In addition, the technology canprovide new functionality and reduce overall optical communication cost thanks to smallerdevices, higher bandwidth and low loss device characteristics that eliminate the need foramplification. The ultimate goal is to create three-dimensional photonic structures that maylead to progressively optical and one day all-optical computing.

4 Introduction

A very recent overview on modelling in photonics, including discussion of importantcomputational methods, was written by Obayya [9].

1.2.3 Overview of commercial software for photonics

An outstanding progress in photonics has been to a great extent possible due to the avail-ability of reliable software. Some of the main commercial players are listed below:

1. At the time of writing, Optiwave (www.optiwave.com) provides comprehensive engi-neering design tools which benefit photonic, bio-photonic and system design engineerswith a comprehensive design environment. Current Optiwave products include twogroups: system and amplifier design, and component design.

For system and amplifier design they offer two packages:• OptiSystem – suite for design of amplifier and optical communication systems.• OptiSPICE – the first optoelectronic circuit design software.

For component design, they have the following products:• OptiBPM, based on the beam propagation method (BPM), offers design of complex

optical waveguides which perform guiding, coupling, switching, splitting, multiplex-ing and demultiplexing of optical signals in photonic devices.

• OptiFDTD, which is based on the finite-difference time-domain (FDTD) algorithmwith second-order numerical accuracy and the most advanced boundary condition,namely uniaxial perfectly matched layer. Solutions for both electric and magneticfields in temporal and spatial domain are obtained using the full-vector differentialform of Maxwell’s equations.

• OptiFiber, which uses numerical mode solvers and other models specialized to fibresfor calculating dispersion, losses, birefringence and polarization mode dispersion(PMD).

• OptiGrating, which uses the coupled mode theory to model the light and enableanalysis and synthesis of gratings.

2. RSoft (www.rsoftdesign.com) product family includes:• Component Design Suite to analyse complex photonic devices and components

through industry-leading computer-aided design,• System Simulation to determine the performance of optical telecom and datacom

links through comprehensive simulation techniques and component models, and• Network Modeling for cost-effective deployment of DWDM and SONET technologies

while designing and optimizing an optical network.3. Photon Design (www.photond.com)

They offer several products for both passive and active component design, such asFIMMWAVE and CrystalWave.

FIMMWAVE is a generic full-vectorial mode finder for waveguide structures. It com-bines methods based on semi-analytical techniques with other more numerical methodssuch as finite difference or finite element.

FIMMWAVE comes with a range of user-friendly visual tools for designingwaveguides, each optimized for a different geometry: rectangular geometries often

5 Optical fibre communication

encountered in epitaxially grown integrated optics, circular geometries for the designof fibre waveguides, and more general geometries to cover, e.g. diffused waveguides orother unusual structures.

CrystalWave is a design environment for the layout and design of integrated opticscomponents optimized for the design of photonic crystal structures. It is based on bothFDTD and finite-element frequency-domain (FEFD) simulators and includes a mask filegenerator optimized for planar photonic crystal structures.

4. CST MICROWAVE STUDIO (www.cst.com/Content/Products/MWS/Overview.aspx)is a specialist tool for the 3D electromagnetic simulation of high frequency components.It enables the fast and accurate analysis of high frequency devices such as antennas,filters, couplers, planar and multi-layer structures.

There are several web resources devoted to photonics software and numerical modellingin photonics. We mention the following: Optical Waveguides: Numerical Modeling Website[10], Photonics resources page [11] and Photonics software [12].

1.3 Optical fibre communication

In this section we will outline the important role played by photonics in the communicationover an optical fibre. Other sub-fields will be summarized in due course.

1.3.1 Short story of optical fibre communication

Light was used as a mean for communication probably since the origin of our civilization.Its modern applications in telecommunication, as developed in the twentieth century, canbe traced to two main facts:

• availability of good optical fibre, and• compact and reliable light sources.

Later in this section, we will discuss main developments in some detail. We start witha short history of communication with the emphasis on light communication. A recentpopular history of fibre optics written for a broad audience was published by Hecht [13].As an introductory work we recommend simple introduction to fundamentals of digitalcommunications by Bateman [14].

1.3.2 Short history of communication

After discovery of electromagnetic waves due to work by Marconi, Tesla and many others,the radio was created operating in the 0.5–2 MHz frequency range with a bandwidth of15 kHz. With the appearance of television which required bandwidth of about 6 MHz,the carrier frequencies moved to around 100 MHz. During the Second World War the

6 Introduction

Laser drivecircuitry

Amplifier andsignal processing

Transmitter Channel Receiver

Optical fibre

Laser Photodetector

Fig. 1.1 Block diagram of a fibre system.

invention of radar pushed the frequencies to the microwave region (around GHz). Thosefrequencies (range of 2.4–5 GHz) are now actively used by cell phones as well as wirelesslinks.

The big step toward higher frequencies was made possible in the 1960s with the inventionof laser. The first one operated at the 694 nm wavelength. This corresponds to carrierfrequency of approximately 5 × 1014 Hz (5 × 105 GHz = 500 THz). Utilization of only1% of that frequency represents a signal channel of about 5 THz which can accommodateapproximately 106 analogue video channels, each with 6 MHz bandwidth or 109 telephonecalls at 5 kHz per call.

A typical communication channel which links two points, point-to-point link (as anexample, see Fig. 1.1 for a generic fibre optics link), consists of an optical transmitterwhich is usually a semiconductor laser diode, an optical fibre intended to carry light beamand a receiver. A semiconductor laser diode can be directly modulated or an externalmodulator is used which produces modulated light beam which propagates in the opticalfibre. Information is imposed into optical pulses. Then the light is input into the opticalfibre where it propagates over some distance, and then it is converted into a signal which ahuman can understand.

All of those elements should operate efficiently at the corresponding carrier’s frequencies(or wavelengths). In the 1960s only one element was in place: a transmitter (laser). Theother two elements were nonexistent.

The next main step was proposed in 1966 by Charles Kao and George Hockham [15] whodemonstrated the first silica-based optical fibre with sufficiently low enough propagationloss to enable its use as a communication medium. Soon, silica-based optical fibre becamethe preferred means of transmission in both long- and short-haul telecommunication net-works. An all-optical network has the potential for a much higher data rate than combinedelectrical and optical networks and allows simultaneous transmission of multiple signalsalong one optical fibre link.

Below, we summarize selected developments of the early history of long-distance com-munication systems (after Hecht [13] and Einarsson [16]):

• TAT-1 (1956) First transatlantic telephone system contained two separate coaxial cables,one for each direction of transmission. The repeaters (based on vacuum tubes) werespliced into the cables at spacing of 70 km. Capacity of the transmission was 36 two-wayvoices.


• TAT-6 (1976) Capacity of 4200 two-way telephone channels. Repeaters (based on tran-sistors) were separated at a distance of 9.4 km.

• TAT-7 (1983) System similar to TAT-6.• TAT-8 (1988) First intercontinental optical fibre system between USA and Europe.• HAW-4 (1988) System installed between USA and Hawaii; similar to TAT-8.• TCP-3 Extension of HAW-4 to Japan and Guam.• TAT-12 (1996) First transatlantic system in service with optical amplifiers.

A typical optical fibre used in TAT-8 system supported single-mode transmission [16].It had outer diameter of 125 µm and core diameter of 8.3 µm. It operated at 1.3 µmwavelength. Digital information was transmitted at 295.6 Mbits. The repeaters whichregenerated optical signals were spaced 46 km apart.

Some of the main developments that took place in relation to development of fibrecommunication are summarized below (compiled using information from Ref. [13]):

• October 1956: L. E. Curtiss and C. W. Peters described plastic-clad fibre at OpticalSociety of America meeting in Lake Placid, New York.

• Autumn 1965: C. K. Kao concludes that the fundamental limit on glass transparency isbelow 20 dB km−1 which would be practical for communication.

• Summer 1970: R. D. Maurer, D. Keck and P. Schultz from Corning Glass Works, USAmake a single-mode fibre with loss of 16 dB km−1 at 633 nm by doping titanium intofibre core.

• April 22, 1977: General Telephone and Electronics. First live telephone traffic throughfibre optics, at 6 Mbits−1 in Long Beach, CA.

• 1981: British Telecom transmits 140 Mbits s−1 through 49 km of single-mode fibre at1.3 µm wavelength.

• Late 1981: Canada begins trial of fibre optics to homes in Elie, Manitoba.• 1988: L. Mollenauer of Bell Labs demonstrates soliton transmission through 4000 km of

single-mode fibre.• February 1991: M. Nakazawa, NTT Japan sends soliton signals through a million km of

fibre.• 1994: World Wide Web grows from 500 to 10 000 servers.• 1996: Introduction of a commercial wavelength-division multiplexing (WDM) system.• July 2000: Peak of telecom bubble. JDS Uniphase announces plans to merge with SDL

Inc. in stock deal valued at $41 billion.• 2001: NEC Corporation and Alcatel transmitted 10 terabits per second through a single

fibre [17].• December 2001: Failure of TAT-8 submarine cable.

The capacity of optical fibre communication systems grew very fast. Comparison withMoore’s law for computers which states that computer power doubles every 18 months,indicates that fibre capacity grows faster [18]. In 1980, a typical fibre could carry about45 Mb s−1; in 2002 a single fibre was able to transmit more than 3.5 Tb s−1 of data. Overthose 22 years the computational power of computers has increased 26 000 times whereasthe fibre capacity increased 110 000 times over the same period of time.

8 Introduction

Egyptian

Venetian

Opt

ical

loss

(dB

/km

)Opticalfibre

1000 1900 1966 1979 1983

102

103

104

105

10

0.1

1.0

106

107

B.C. A.D.

3000

Fig. 1.2 Historical reduction of loss as a function of time from ancient times to the present. Copyright 1989 IEEE. Reprinted,with permission from S. R. Nagel, IEEE Communications Magazine, 25, 33 (1987).

1.3.3 Development of optical fibre

Glass is the principal material used to fabricate optical fibre. To be useful as a transmissionmedium it has to be extremely clean and uniform. In Fig. 1.2 we illustrated the history ofreduction losses in glass as a function of time (after [19]). To make practical use of glass asa medium used to send light, an important element, and in fact the very first challenge wasto develop glass so pure that 1% of light entering at one end would be retained at the otherend of a 1 km-thick block of glass. In terms of attenuation this 1% corresponds to about 20decibels per kilometre (20 dB km−1).

In 1966 Kao and Hockhman [15] suggested that the high loss in glass was due toimpurities and was not the intrinsic property of glass. In fact, Kao suggested to use opticalfibre as a transmission medium.

Kao’s ideas were made possible in 1970 with the fabrication of low-loss glass fibre bythe research group from Corning Glass Works [20]. That work established a potential ofoptical communications for providing a high bandwidth and long distance data transmissionnetwork. Corning Glass Works were the first to produce an optical glass with a transmissionloss of 20 dB km−1, which was thought to be the threshold value to permit efficient opticalcommunication.


wavelength (µm)

loss

(dB

/km

)1.61.41.21.0

0.1

0.5

510

50100

1

0.8

Firstwindow

Thirdwindow

Secondwindow

Fig. 1.3 Spectral loss of a typical low-loss optical fibre. Adapted from [22] with permission from the Institution of Engineeringand Technology.

Before the Corning work, around 1966, the power loss for the best available glass wasabout 1000 dB km−1 which implies a 50% power loss in a distance of about 3 m [21]. Today,most optical fibres in use are so-called single-mode fibre able to maintain single mode ofoperation (see Chapter 5 for the relevant definitions). Commercially available fibres havelosses of about 0.2 dB km−1 at the wavelength of 1550 nm and are capable of transmittingdata at 2–10 Gbit s−1.

The measured spectral loss of a typical low-loss optical fibre is shown in Fig. 1.3 [22].The measured fibre had 9.4 µm core diameter and a 0.0028 refractive index differencebetween the core and cladding. Peaks were found at about 0.94, 1.24 and 1.39 µm. Theywere caused by OH vibrational absorption. The rise of the loss from 1.7 µm is mainlyattributed to the intrinsic infrared absorption of the glass.

The locations of the first, second and third communications windows are also shown. Themost common wavelengths used for optical communication span between 0.83–1.55 µm.Early technologies used the 0.8–0.9 µm wavelength band (referred to as the first window)mostly because optical sources and photodetectors were available at these wavelengths.Second telecommunication window, centered at around 1.3 µm, corresponds to zero dis-persion of fibre. Dispersion is a consequence of the velocity propagation being different fordifferent wavelengths of light. As a result, when pulse travels through a dispersive media ittends to spread out in time, the effect which ultimately limits speed of digital transmission.The third window, centered at around 1.55 µm, corresponds to the lowest loss. At thatwavelength, the glass losses approach minimum of about 0.15 dB km−1.

For the sake of comparison, the loss of about 0.2 dB km−1 corresponds to 50% powerloss after propagation distance of about 15 km.

1.3.4 Comparison with electrical transmission

Historically, copper was a traditional medium used to transmit information by electri-cal means. Its data handling capacity is, however, limited. Only around 1970, after the

10 Introduction

fabrication of relatively low-loss optical fibre, it started to be used in an increasing scale.From practical perspective, the choice between copper or glass is based on several factors.

In general, optical fibre is chosen for systems which are used for longer distances andhigher bandwidths, whereas electrical transmission is preferred in short distances andrelatively low bandwidth applications. The main advantages of electrical transmission areassociated with:

• relatively low cost of transmitters and receivers• ability to carry electrical signals and also electrical power• for not too large quantities, relatively low cost of materials• simplicity of connecting electrical wires.

However, it has to be realized that optical fibre is much lighter than copper. For example,700 km of telecommunication copper cable weighs 20 tonnes, whereas optical fibre cableof the same length weighs only around 7 kg [23].

The main advantages of optical fibre are:

• fibre cables experience effectively no crosstalk• they are immune to electromagnetic interference• lighter weight• no electromagnetical radiation• small cable size.

1.3.5 Governing standards

In order for various manufacturers to be able to develop components that function com-patibly in fibre optic communication systems, a number of standards have been developedand published by the International Telecommunications Union (ITU) [24]. Other standards,produced by a variety of standards organizations, specify performance criteria for fibre,transmitters and receivers to be used together in conforming systems.

These and related standards are covered in a recent publication [25]. It is the workof over 25 world-renowned contributors and editors. It provides a reference to over ahundred standards and industry technical specifications for optical networks at all levels:from components to networking systems through global networks, as well as coverage ofnetworks management and services.

The ITU has specified six transmission bands for fibre optic transmission [18]:

1. the O-band (original band) – 1260–1310 nm,2. the E-band (extended band) – 1360–1460 nm,3. the S-band (short band) – 1460–1530 nm,4. the C-band (conventional band) – 1530–1565 nm,5. the L-band (long band) – 1565–1625 nm,6. the U-band (ultra band) – 1625–1675 nm.

A seventh band, not defined by the ITU, runs around 850 nm and is used in private networks.


Time-division multiplexing (TDM)

Wavelength-division multiplexing (WDM)

Channels

Channels

I

I

t

1

1

12

2

26

6

68

8

λ

87

7

75

5

54

4

43

3

3

Fig. 1.4 Illustration of TDM andWDM formats.

MUX DMUX1

11 λλ λ λ

λ

λ

λ

λ

λ

22 2

3

3 3

Fig. 1.5 WDM transmission system.

1.3.6 Wavelength division multiplexing (WDM)

Wavelength division multiplexing (WDM) is a technique which allows optical signals withdifferent wavelengths to propagate in a single optical fibre without affecting one another.The technique was invented to increase the information-carrying capacity of a fibre. WDMis not the only possible way to increase fibre capacity. In Fig. 1.4 we illustrate two basicformats, each allowing an increase of fibre capacity, time-division multiplexing (TDM) andwavelength-division multiplexing (WDM). In both formats messages from various channelsare combined (multiplexed) onto a single information channel. This can be done in time orat various frequencies (wavelengths).

There is extensive literature on the principles, concepts and components for WDM,see [26], [27], [28], [29], [30] to just name a few. The WDM transmission system isshown in Fig. 1.5. Here, MUX denotes a multiplexer. Its role is to combine light fromseveral sources, each operating at a particular wavelength carrying one channel, to thetransmission fibre. At the receiving end, the demultiplexer (DMUX) separates differentwavelengths.

12 Introduction

1.3.7 Solitons

In the mid 1980s researchers started experimenting with solitons. The soliton phenomenonwas first described by John Scott Russell, who in 1834 observed a solitary wave in a narrowwater channel (the Union Canal in Scotland).

Solitons are pulses of special shape which travel without changing it. Pulse broadening isdue to the fact that some wavelengths travel faster than other wavelengths, the effect knownas dispersion. In fibre, there also exists nonlinearity. When taken together, dispersion andnonlinearity, they compensate for each other, thus producing conditions where pulse shaperemains unchanged. To form a soliton, the initial pulse must have a particular peak energyand pulse shape.

In 1973, Akira Hasegawa of AT&T Bell Labs was the first to suggest that solitons couldexist in optical fibre. In 1988, Linn Mollenauer and his team transmitted soliton pulses over4000 km [31]. A recent overview of soliton-based optical communications was providedby Hasegawa [32] and by Mollenauer and Gordon [33].

1.4 Biological andmedical photonics

A special issue of the IEEE Journal of Selected Topics in Quantum Electronics [34]summarizes hot topics in bio-photonics until circa 2005. Some of them are directlyrelated to computational bio-photonics, such as elastic light scattering properties oftissues [35].

Two special issues of IEEE Journal of Selected Topics in Quantum Electronics on bio-photonics [36], [37] summarize recent progress and trends. Biological applications ofstimulated parametric emission microscopy and stimulated Raman scattering microscopyare reviewed by Kajiyama et al. [38]. As an introduction to the subject we recommend thebook by Prasad [39].

1.5 Photonic sensors

Photonic sensors have been the subject of intensive research over the last two decades. Theyare emerging as an important branch of photonics. Their development is based on analysingdifferent components of the optical signal, like intensity, polarization, pulse shape or arrivaltime and also other phenomena like interference. Utilizing those various possibilities leadsto different types of sensors. There are almost endless possibilities for their use. The bestknown applications are in civil and military environments for detection of a wide varietyof biological, chemical and nuclear agents.

An important medium for sensing is the optical fibre and its recent variations whichfound their own ways of applications in sensing. These include: photonic band gapfibres (PBG), microstructure optical fibres (MOF), random hole optical fibres (RHOF)

13 Silicon photonics

and hybrid ordered random hole optical fibres (HORHOF). They offer higher resolu-tion, lower cost and/or expanded detection range capability for sources and detectionschemes.

In this book we will not explicitly concentrate on analysis and/or simulations of photonicsensors, although some of the discussed properties of photonic devices can be directlyapplied in sensors.

As an example, the use of light emitting diodes (LED) in sensing has been summarized re-cently by O’Toole and Diamond [40]. The authors described development and advancementof LED-based chemical sensors and sensing devices.

Another example is device based on a high-finesse, whispering-gallery-mode disk res-onator that can be used for the detection of biological pathogens [41]. The device operatesby means of monitoring the change in transfer characteristics of the disk resonator whenbiological materials fall onto its active area.

The open access journal Sensors (ISSN 1424-8220; CODEN: SENSC9) [42] on scienceand technology of sensors and biosensors is published monthly online. A special issue:Photonic Sensors for Chemical, Biological, and Nuclear Agent Detection was publishedrecently. It can be considered a good source of current developments and applications ofnew physical phenomena in sensing.

1.6 Silicon photonics

The application of silicon in photonics certainly needs a separate discussion. Silicon is amaterial of choice in electronics and it is also economical to build photonic devices onthis material [43], see also the contribution by Soref [44] for an overview of optoelectronicintegrated circuits. However, silicon is not a direct-gap material and therefore cannotefficiently generate light. Intensive work on those issues is taking place in many institutions,see the recent summary of the Intel-UC Santa Barbara collaboration on the hybrid siliconlaser [45].

On another front, associated with computer technology, the continuation of Moore’s Lawand progress which it reflects is becoming increasingly dependent on ultra-fast data transferbetween and within microprocessor. The existing electrical interconnects are expected tobe replaced by high speed optical interconnects. They are seen as a promising way forward,and silicon photonics is seen as particularly useful, due to the ability to integrate electronicand optical components on the same silicon chip.

Particularly significant progress has been seen in the advances of optical modulatorsbased on silicon, see the recent report by a leading collaboration [46].

In designing and optimizing those devices simulations play a very important role. Re-cent advances in modelling and simulation of silicon photonic devices are summarizedby Passaro and De Leonardis [47]. They review recently developed simulation methodsfor submicrometre innovative Silicon-on-Insulator (SOI) guiding structures and photonicdevices.

14 Introduction

1.7 Photonic quantum information science

A new emerging direction within photonics is related to quantum communications andquantum information processing. Recent progress in this subfield is summarized by Politiet al. [48]. Here, we briefly outline recent main developments.

Photons potentially can play the crucial role in the quantum information processing dueto their low-noise properties and ease of manipulation at the single qubit level. In recentyears several quantum gates have been implemented using integrated photonics circuits.

1. A theoretical proposal for the implementation of the quantum Hadamard gate usingphotonic crystal structures [49]. An integrated optics Y-junction beam splitter has beenproposed for realization of this gate. Reported numerical simulations support theoreticalconcepts.

2. A group from Bristol demonstrated the capability to build photonic quantum circuitsin miniature waveguide devices [50] and optical fibres [51]. They reported recently on high-fidelity silica-on-silicon integrated optical realizations of key quantum photonic circuits,including two-photon quantum interference and a controlled-NOT logic gate [52]. Theyhave also demonstrated controlled manipulation of up to four photons on-chip, includinghigh-fidelity single qubit operations, using a lithographically patterned resistive phaseshifter [53].

References

[1] Photonics West, http://spie.org/x2584.xml 3 July 2012.[2] OFC Conference, www.ofcnfoec.org/home.aspx, 3 July 2012.[3] Photonics Global Conference, www.photonicsglobal.org/index.php, 3 July 2012.[4] C. R. Pollock and M. Lipson. Integrated Photonics. Kluwer Academic Publishers,

Boston, 2003.[5] J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade. Photonic Crystals:

Molding the Flow of Light. Princeton University Press, Princeton and Oxford, 2008.[6] J.-M. Jin. Theory and Computation of Electromagnetic Fields. Wiley-IEEE Press,

Hoboken, NJ, 2010.[7] A. Taflove and S. C. Hagness. Computational Electrodynamics. The Finite-Difference

Time-Domain Method. Artech House, Boston, 2005.[8] D. Gallagher. Photonic CAD matures. IEEE LEOS Newsletter, 2:8–14, 2008.[9] S. Obayya. Computational Photonics. Wiley, Chichester, 2011.

[10] Optical Waveguide Modeling, http://optical-waveguides-modeling.net/, 3 July 2012.[11] Photonics resources page, www.ashokpm.org/photonics.htm, 3 July 2012.[12] Photonics software, www.skywise711.com/lasers/software.html, 3 July 2012.[13] J. Hecht. City of Light: The Story of Fiber Optics. Oxford University Press, Oxford,

1999.

15 References

[14] A. Bateman. Digital Communications: Design for the Real World. Addison-Wesley,Harlow, 1999.

[15] K. C. Kao and G. A. Hockman. Dielectric-fibre surface waveguides for optical fre-quencies. Proc. IEE, 133:1151–8, 1966.

[16] G. Einarsson. Principles of Lightwave Communications. Wiley, Chichester, 1996.[17] J. Hecht. Computers full of light: a short history of optical data communications. In

C. DeCusatis, ed., Handbook of Fiber Optic Data Communication, pp. 3–17. Elsevier,Amsterdam, 2008.

[18] D. R. Goff, K. S. Hansen, and M. K. Stull. Fiber Optic Video Transmission: TheComplete Guide. Focal Press, Oxford, 2002.

[19] S. R. Nagel. Optical fiber – the expanding medium. IEEE Communications Magazine,25:33–43, 1987.

[20] F. P. Kapron, D. B. Keck, and R. D. Maurer. Radiation losses in glass optical waveg-uides. Appl. Phys. Lett., 17:423–5, 1970.

[21] G. Ghatak and K. Thyagarajan. Introduction to Fiber Optics. Cambridge UniversityPress, Cambridge, 1998.

[22] T. Miya, Y. Terunuma, T. Hosaka, and T. Miyashita. Ultimate low-loss single-modefibre at 1.55 µm. Electronics Letters, 15:106–8, 1979.

[23] Optical fiber, http://en.wikipedia.org/wiki/Optical fiber, 3 July 2012.[24] International Telecommunication Union, www.itu.int/en/pages/default.aspx, 3 July

2012.[25] K. Kazi. Optical Networking Standards: A Comprehensive Guide. Springer, New

York, 2006.[26] R. T. Chen and L. S. Lome, eds., Wavelength Division Multiplexing. SPIE Optical

Engineering Press, Bellington, WA, 1999.[27] G. Keiser. Optical Fiber Communications. Third Edition. McGraw-Hill, Boston, 2000.[28] S. V. Kartalopoulos. DWDM. Networks, Devices and Technology. IEEE Press, Wiley,

Hoboken, NJ, 2003.[29] J. Zheng and H. T. Mouftah. Optical WDM Networks. Concepts and Design Principles.

IEEE Press, Wiley, Hoboken, NJ, 2004.[30] J. C. Palais. Fiber Optic Communications. Fifth Edition. Prentice Hall, Upper Saddle

River, NJ, 2005.[31] L. F. Mollenauer and K. Smith. Demonstration of soliton transmission over more

than 4000 km in fiber with loss periodically compensated by Raman gain. Opt. Lett.,13:675–7, 1988.

[32] A. Hasegawa. Soliton-based optical communications: An overview. IEEE J. Select.Topics Quantum Electron., 6:1161–72, 2000.

[33] L. F. Mollenauer and J. P. Gordon. Solitons in Optical Fibers. Fundamentals andApplications. Elsevier, Amsterdam, 2006.

[34] R. R. Jacobs. Introduction to the issue on biophotonics. IEEE J. Select. Topics Quan-tum Electron., 11:729–32, 2005.

[35] X. Li, A. Taflove, and V. Backman. Recent progress in exact and reduced-ordermodeling of light-scattering properties of complex structures. IEEE J. Select. TopicsQuantum Electron., 11:759–65, 2005.

16 Introduction

[36] I. K. Ilev, L. V. Wang, S. A. Boppart, S. Andersson-Engels, and B.-M. Kim. Intro-duction to the special issue on biophotonics. Part 1. IEEE J. Select. Topics QuantumElectron., 16:475–7, 2010.

[37] I. K. Ilev, L. V. Wang, S. A. Boppart, S. Andersson-Engels, and B.-M. Kim. Intro-duction to the special issue on biophotonics. Part 2. IEEE J. Select. Topics QuantumElectron., 16:703–5, 2010.

[38] S. Kajiyama, Y. Ozeki, K. Fukui, and K. Itoh. Biological applications of stimulatedparametric emission microscopy and stimulated Raman scattering microscopy. InB. Javidi and T. Fournel, eds., Information Optics and Photonics. Algorithms, Systems,and Applications, pp. 89–100. Springer, New York, 2010.

[39] P. N. Prasad. Introduction to Biophotonics. Wiley, Hoboken, NJ, 2003.[40] M. O’Toole and D. Diamond. Absorbance based light emitting diode optical sensors

and sensing devices. Sensors, 8:2453–79, 2008.[41] R. W. Boyd and J. E. Heebner. Sensitive disk resonator photonic biosensor. Applied

Optics, 40:5742–7, 2001.[42] Sensors – Open Access Journal, www.mdpi.com/journal/sensors, 3 July 2012.[43] G. T. Reed and A. P. Knights. Silicon Photonics: An Introduction. Wiley, Chichester,

2004.[44] R. Soref. Introduction: the opto-electronics integrated circuit. In G. T. Reed, ed.,

Silicon Photonics: The State of the Art, pp. 1–13. Wiley, Hoboken, NJ, 2008.[45] J. E. Bowers, D. Liang, A. W. Fang, H. Park, R. Jones, and M. J. Paniccia. Hybrid

silicon lasers: The final frontier to integrated computing. Optics & Photonics News,21:28–33, 2010.

[46] D. J. Thomson, F. Y. Gardes, G. T. Reed, F. Milesi, and J.-M. Fedeli. High speed siliconoptical modulator with self aligned fabrication process. Optics Express, 18:19064–9,2011.

[47] V. M. N. Passaro and F. De Leonardis. Recent advances in modelling and simulationof silicon photonic devices. In G. Petrone and G. Cammarata, eds., Modelling andSimulation, pp. 367–90. I-Tech Education and Publishing, Vienna, 2008.

[48] A. Politi, J. Matthews, M. G. Thompson, and J. L. O’Brien. Integrated quantumphotonics. IEEE J. Select. Topics Quantum Electron., 15:1673–84, 2009.

[49] S. Salemian and S. Mohammadnejad. Quantum Hadamard gate implementation usingplanar lightwave circuit and photonic crystal structures. American Journal of AppliedSciences, 5:1144–8, 2008.

[50] A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien. Silica-on-silicon waveguidequantum circuits. Science, 320:646–9, 2008.

[51] A. S. Clark, J. Fulconis, J. G. Rarity, W. J. Wadsworth, and J. L. O’Brien. All-optical-fiber polarization-based quantum logic gate. Phys. Rev. A, 79:030303, 2009.

[52] A. Laing, A. Peruzzo, A. Politi, M. Rodas Verde, M. Halder, T. C. Ralph, M. G.Thompson, and J. L. O’Brien. High-fidelity operation of quantum photonic circuits.Appl. Phys. Lett., 97:211109, 2010.

[53] J. C. F. Matthews, A. Politi, A. Stefanov, and J. L. O’Brien. Manipulation of mul-tiphoton entanglement in waveguide quantum circuits. Nature Photonics, 3:346–50,2009.

2 Basic facts about optics

In this chapter, we review basic facts about geometrical optics. Geometrical optics isbased on the concept of rays which represent optical effects geometrically. For an excellentintroduction to this topic consult work by Pedrotti and Pedrotti [1].

2.1 Geometrical optics

2.1.1 Ray theory and applications

We visualize propagation of light by introducing the concept of rays. These rays obeyseveral rules:

• In a medium they travel with velocity v given by

v = c

n(2.1)

where c = 3 × 108 m/s is the velocity of light in a vacuum and the quantity n is knownas refractive index. (We introduced here the notation n for refractive index instead ofwidely accepted n in the optics literature to avoid confusion with electron density, nwhich will be used in later chapters.) Some typical values of refractive indices are shownin Table 2.1.

• In a uniform medium rays travel in a straight path.• At the plane of interface between two media, a ray is reflected at the angle equal to the

angle of incidence, see Fig. 2.1. At the plane interface, the direction of transmitted ray isdetermined by Snell’s law

n1 sin θ1 = n2 sin θ2 (2.2)

where n1 and n2 are the values of refractive indices in both media and θ1 and θ2 are theangles of incidence and refraction, respectively.

The general relation between incoming ray A and the reflected ray B is

B = R · A (2.3)

with R reflection coefficient (complex). Reflection coefficients for two main types of waveswhich can propagate in planar waveguides are given by Fresnel formulas and will bedetermined in the next chapter.

17


Table 2.1 Refractive index for some materials.

Material Refractive index

air 1.0water 1.33silica glass 1.5GaAs 3.35silicon 3.5germanium 4.0

θ2

A B

C

interface

n2

n1 θ1 θ1

Fig. 2.1 Snell’s law.

n2

n1

Θc

Fig. 2.2 Illustration of critical angle.

2.1.2 Critical angle

From Snell’s law one can deduce the conditions under which light ray will stay in thesame medium (e.g. dense) after reflection, see Fig. 2.2. The situation shown correspondsto θ2 = π/2. Angle θ1 then becomes known as the critical angle, θc and its value is:

sin θc = n2

n1(2.4)

19 Geometrical optics

Fibre

f

Fig. 2.3 Focusing light beam onto a fibre.

f

lensaxis

focal point

Fig. 2.4 Focusing an off-axis beam.

2.1.3 Lenses

We discuss here only thin lenses, defined when the vertical translation of a ray passingthrough lens is negligible. The ray enters and leaves the lens at approximately the samedistance from the lens axis.

An important use of lenses in fibre optics is to focus light onto a fibre, see Fig. 2.3. Here,a collimated beam of light (parallel to the lens axis) is focused to a point. The point isknown as the focal point and is located at a distance f , known as focal length from the lens.All the collimated rays converge to the focal point.

Focal distance f is related to the curvatures R1 and R2 of the spheres forming lens as

1

f= (n − 1)

(1

R1+ 1

R2

)(2.5)

If beam of light travels at some angle relative to the lens axis, Fig. 2.4, the rays will befocused in the focal plane.


n(r)

r

n maxA B

C ED

F

fd

Fig. 2.5 A cylindrical disk with graded refractive index.

2.1.4 GRIN systems

GRIN systems whose name originates from GRadient in the INdex of refraction play therole similar to lenses. Here, the refractive index of the material is inhomogeneous. Spatialdependence of refractive index serves as yet another parameter in the design of an ideallens.

The simple analysis of GRIN structure now follows [2], see Fig. 2.5. We have a flatcylindrical disk where the refractive index n(r) depends on radius r (distance from theaxis), keeps its maximum at the optical axis and decreases outwards. In order to focusoptical rays at the focal point F , the optical paths of all rays must be the same. (Optical pathis defined as a product of geometrical distance in a given region and the value of refractiveindex in that region.) Optical path difference of the central ray and the one going throughregion with n(r) between planar wave front on the left and spherical wavefront on the rightshould be zero

nmax · d ≈ n(r) · d + DE (2.6)

Distance DE is determined as follows: EF = f , where f is the focal length, DF = DE + fand DF ≈

√r2 + f 2. Combining those relations, we have

DE =√

r2 + f 2 − f (2.7)

Use the expansion √r2 + f 2 = f

√1 + r2

f 2≈ f

(1 + r2

2 f 2

)(2.8)

Apply the above expansion in (2.7) and obtain

DE ≈ f + r2

2 f 2− f = r2

2 f 2(2.9)

Finally, the dependence of refractive index is

n(r) = nmax − r2

2df 2(2.10)

21 Wave optics

The above equation tells us that in order to focus parallel light by the GRIN plate, the indexof refraction should depend parabolically on radius.

2.2 Wave optics

Light is an electromagnetic wave and consists of an electric field vector and a perpendicularmagnetic field vector oscillating at a very high rate (of the order of 1014 Hz) and travellingin space along direction which is perpendicular to both electric and magnetic field vectors.(These topics are discussed in more detail in the next chapter.) Here we state that the electricfield vector E(r,t) obeys the following wave equation:

∇2E(r, t) − 1

c2

∂2E(r, t)

∂t2= 0 (2.11)

The above wave equation is time invariant; that is it does not change under the replacementt → −t. We will soon utilize that fact in establishing the so-called Stokes relations.

The one-dimensional wave equation in a medium is

1

v2

∂2

∂t2E(z, t) = ∂2

∂z2E(z, t)

where v is the velocity of light in the medium. An exact solution can be expressed in termsof two waves E+ and E− travelling in the positive and negative z-directions

E(z, t) = E+(z − vt) + E−(z + vt)

Dispersion relation for the above wave equation is obtained by substituting E(z, t) =E0 sin (ωt − kz) into the wave equation, performing differentiation and dividing both sidesby E(z, t). The obtained dispersion relation which relates k and ω is

ω = ±v · k

It shows a linear dependence between angular frequency ω of the propagating wave andits wavenumber k. Angular frequency ω is related to frequency as ω = 2π f whereaswavenumber k is determined as

k = ω

v= 2π

λ

Here λ is the wavelength of the light wave and v is the phase velocity of the wave. Thefactor ωt − kz is known as the phase of the wave.

Example When electromagnetic wave propagates in a medium where losses are significant,the attenuation must be incorporated into an expression for electric field of the wave. Therelevant expression is

E(z, t) = E0e−αz sin(ωt − kz)


where α is the attenuation coefficient. The intensity of light is proportional to the squareof its electric field. Therefore, the power of the light beam decreases as exp(-2αL) duringthe propagation distance L. If we measure the distance in km, the unit of α is km−1. Powerreduction in decibels (dB) is determined as

dB = 10 log10 e−2αL (2.12)

Find the relation between the attenuation coefficient and the power change in dB/km.

SolutionOne has

Loss(dB) = 10 log10 e−2αL

= −20 · α · L · log e

= −8.685 · α · L

Therefore

Loss(dB)/L(km) = −8.685 · α

or

dB/km = −8.685 · α (2.13)

where α is in km−1.

2.2.1 Phase velocity

Consider a plane monochromatic wave propagating along the z-axis in an infinite medium,see Fig. 2.6

E(z, t) = E0cos (kz − ωt) (2.14)

Select one point, say at the amplitude crest, and analyse its movement. If we assume thatthis point maintains constant phase as

kz − ωt = const

we can determine velocity of that point. From the above relation

kdz

dt− ω = 0

or

vp = dz

dt= ω

k(2.15)

The quantity vp is known as phase velocity and it represents the velocity at which a surfaceof constant phase moves in the medium. Velocity in Eq. (2.1) should be interpreted as phasevelocity and Eq. (2.1) be replaced by

vp = c

n(2.16)

23 Wave optics

0 0.5 1 1.5 2 2.5 3

× 10−14

−1.5

−1

−0.5

0

0.5

1

1.5

2

Time

Am

plitu

de

z(t)

Fig. 2.6 Illustration of phase velocity.

2.2.2 Group velocity

The group velocity describes the speed of propagation of a pulse in a medium. To seethis fact, consider propagation of two plane waves with different parameters but equalamplitudes

E1(z, t) = E0 cos (k1z − ω1t) , E2(z, t) = E0 cos (k2z − ω2t) (2.17)

Assume that their frequencies and wavenumbers differ by 2�ω and 2�k and write

ω1 = ω + �ω, ω2 = ω − �ω

k1 = k + �k, k2 = k − �k

Superposition of those waves results in a wave E(z, t) which is

E(z, t) = E1(z, t) + E2(z, t)

= E0 {cos [(k + �k) z − (ω + �ω) t] + cos [(k − �k) z − (ω − �ω) t]}Using trigonometric identity

2 cos α cos β = cos (α + β) + cos (α − β)

allows us to write expression for E(z, t) as

E(z, t) = 2E0 cos (kz − ωt) cos (�kz − �ωt) (2.18)

Eq. (2.18) represents a wave at carrier frequency ω that is modulated by a sinusoidalenvelope at the beat frequency �ω, see Fig. 2.7. Phase velocity of the carrier wave is


0 1 2 3 4× 10−13

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Time

Am

plitu

de

Fig. 2.7 Illustration of group velocity.

vp = ω/k. The envelope moves with velocity vg which is

vg = �ω

�k→ dω

dk(2.19)

In a free space ω = kc which is known as dispersion relation, the group velocity isvg = dω

dk = c. In a free space (vacuum), the phase and group velocities are identical. Inmedia, the dispersion relation, i.e. ω = ω(k) dependence is usually a complicated function.

Example Write a MATLAB program to visualize phase and group velocities based onEqs. 2.15 and 2.19.

SolutionThe MATLAB code for phase velocity is provided in the Listing 2A.1 and in theListing 2A.2 for group velocity. The results are shown in Fig. 2.6 and Fig. 2.7.

2.2.3 Stokes relations

Stokes relations relate reflectivities and transmittivities of light at the plane of interfacebetween two media. Assume that electric field E which is represented by the correspondingray interacts with the interface and is partially reflected and partially transmitted, see Fig. 2.8.Here r, t are reflection and transmission coefficients for fields (not intensities) propagatingfrom medium 2 to medium 1 (here, from top to bottom). Similarly, r′, t ′ denote the reflectionand transmission coefficients for fields propagating from medium 1 to medium 2.

25 Wave optics

2

1

tE

rEE

Fig. 2.8 Transmission and reflection at the interface used in deriving Stokes relations.

2

1

tE

rEE

Fig. 2.9 Transmission and reflection at the interface used in deriving Stokes relations. Time-reversed process.

2

1

tErtE+r'tE

rrE+t'tE rE

Fig. 2.10 Transmission and reflection at the interface used in deriving Stokes relations. Full processes.

At the beginning of this section we have established the property of time invariance ofthe wave equation. Time reversal invariance tells us that the above picture (reflection andtransmission) should also work in reverse time, see Fig. 2.9 where we combined rays tEand rE to form ray E.

However, when one inputs two rays (tE and rE) at the interface between two media 1 and2, one from above and one from below, there should exist partial reflection and transmissionfor both rays, as is illustrated in Fig. 2.10.

A comparison of Fig. 2.9 and Fig. 2.10 reveals physically identical situations, andtherefore the following relations hold:

r2E + t ′tE = E

rtE + r′tE = 0

From the above we obtain Stokes relations

t ′t = 1 − r2

r′ = −r


hA

B

C

D

E

1 23

nf

nΘi

0

Θi

Θt

Fig. 2.11 Interference in a single film with light incident at arbitrary angle θi.

2.2.4 Interference in dielectric film

Interference in dielectric film is an important phenomenon behind operation of opticaldevices, like filters. It is also a fundamental effect for the operation of the Fabry-Perotinterferometer.

To analyse this interference, consider a single layer dielectric film having refractiveindex n f which is in contact with the medium having refractive index n0 from above,see Fig. 2.11. For such a configuration, we will derive conditions for constructive anddestructive interference. We will do this by calculating the optical path difference.

The incident beam 1 creates on reflection and transmission two emerging beams 2 and3. The phase difference between beams 2 and 3 as measured at points C and D is due tooptical path difference � (optical path is defined as a product of geometrical path timesrefractive index) and is

� = n f (AB + BC) − n0AD (2.20)

From Fig. 2.11 one can determine the following trigonometric relations:

AD

AC= cos(90o − θi) = sin θi (2.21)

h

AB= cos θt (2.22)

AE

h= tan θt (2.23)

and also

AC = 2 · AE (2.24)

Combining Eqs. (2.23) and (2.24), one obtains

AC = 2 · t · tan θt (2.25)

Substituting (2.25) into (2.21) we obtain geometrical path for AD part

AD = 2 · h · tan θt · sin θi (2.26)

Geometrical path AB + AC is obtained using Eq. (2.22) as

AB + AC = 2 · AB = 2 · t

cos θt(2.27)

27 Wave optics

h

1 2 3 4E0

nf

n0

Fig. 2.12 Multiple interference in a parallel plate.

Combining Eqs. (2.26) and (2.27) with (2.20) one finds the optical path difference as

� = n f · 2 · t

cos θt− n0 · 2 · t · sin θt

cos θt· sin θi (2.28)

= 2hn f

cos θt

{1 − n0

n fsin θt sin θi

}(2.29)

Using Snell’s law

n0 sin θi = n f sin θt

and the following trigonometric identity

cos2 θt + sin2 θt = 1

the curly bracket in Eq. (2.29) gives

1 − n0

n fsin θt sin θi = 1 − 1

n fsin θtn f sin θt

= 1 − sin2 θt = cos2 θt

Employing the last result in (2.29), we finally obtain for the optical path difference

� = 2 · h · n f · cos θt (2.30)

Thus, the optical path difference is expressed in terms of the angle of refraction. For normalincidence θi = θt = 0 and � = 2n f h. The phase difference δ acquired by light ray due totravel in the film corresponding to optical path difference � is

δ = k · � = 2π

λ0· � (2.31)

2.2.5 Multiple interference in a parallel plate

Based on discussion from the previous section, we will now consider interference of multiplebeams in a parallel plate, see Fig. 2.12. Here, E0 is the amplitude of the incoming light, r, tare the coefficients of external reflections and transmissions, and r′, t ′ are coefficients ofinternal reflections and transmissions.


Phase difference between two successive reflected beams which exists due to travel inthe film having refractive index n f is, Eq. (2.31)

δ = k · �

where � is given by Eq. (2.30). Analysing Fig. 2.12, one can write the following relationsfor outgoing beams

E0eiωt incident beamE1 = rE0eiωt first reflected beamE2 = tt ′r′E0eiωt−iδ second reflected beamE3 = tt ′r′3E0eiωt−i2δ third reflected beamE4 = tt ′r′5E0eiωt−i3δ fourth reflected beam

kth reflected beam is

Ek = tt ′r′(2k−1)E0eiωt−i(k−1)δ

Sum of all reflected beams, ER is

ER =∞∑

k=1

Ek = rE0eiωt +∞∑

k=2

tt ′r′(2k−3)E0eiωt e−i(k−2)δ

= E0eiωt

{r + tt ′r′e−iδ

∞∑k=2

r′(2k−4)e−i(k−2)δ

}

To perform the above infinite sum, use an elementary result for the sum of the geometricseries

S =∞∑

k=2

xk−2 = 1 + x + x2 + · · ·

For |x| < 1, the sum is S = 1/(1 − x). In our case here, x = r′2e−iδ , and total reflection isthus

ER = E0eiωt

{r + tt ′r′e−iδ

1 − r′2e−iδ

}Using Stokes relations, we obtain

ER = E0eiωt

{r − (1 − r2)re−iδ

1 − r2e−iδ

}= E0eiωt r(1 − e−iδ )

1 − r2e−iδ

Light intensity of the reflected beam is proportional to the irradiance IR which is

IR ∼ |ER|2 = E20 r2

{eiωt (1 − e−iδ )

1 − r2e−iδ

}{e−iωt (1 − eiδ )

1 − r2eiδ

}= E2

0 r2 2(1 − cos δ)

1 + r4 − 2r2 cos δ

29 Wave optics

Introduce Ii as the irradiance of the incident beam and IT as the irradiance of the transmittedbeam. We have

IR

Ii= |ER|2

|Ei|2

and

IR + IT = Ii

which originates from energy conservation in nonabsorbing films. Using the above relations,one obtains

IR = 2r2(1 − cos δ)

1 + r4 − 2r2 cos δIi (2.32)

and

IT = (1 − r2)2

1 + r4 − 2r2 cos δIi (2.33)

2.2.6 Fabry-Perot (FP) interferometer

The FP interferometer is formed by two parallel plates with some medium (often air)between them. Expressions for transmission and reflection in such a system have beenderived in the previous section.

The expression for transmission can be used to determine the transmittance T (or Airyfunction) as follows:

T ≡ IT

Ii= (1 − r2)2

1 + r4 − 2r2 cos δ

= 1 − 2r2 + r4

1 + r4 − 2r2 + 4r2 sin2 δ2

= 1

1 + 4r2

(1−r2 )2 sin2 δ2

using trigonometric relation cos δ = 1 − 2 sin2 δ2 .

Define coefficient of finesse

F ≡ 4r2

(1 − r2)2(2.34)

Transmittance is thus expressed as

T = 1

1 + F sin2 δ2

(2.35)


0 5 10 15 200

0.2

0.4

0.6

0.8

1

Tra

nsm

itta

nce

Phase difference (rad)

2 π 4 πr = 0.8

r = 0.4

r = 0.2

Fig. 2.13 Fabry-Perot fringe profile.

Example Write a MATLAB program to analyse the transmittance of an FP etalon given byEq. (2.35).

SolutionThe MATLAB code is shown in Appendix, Listing A 2.3 and the FP fringe profile is shownin Fig. 2.13. The figure shows plots of transmission as a function of phase difference δ forvarious values of external reflection r.

2.3 Problems

1. Use Snell’s law to derive an expression for critical angle θc. Compute the value of θc

for a water-air interface (nwater = 1.33).2. Isotropic light source located at distance d under water illuminates circular area of a

radius 5 m. Determine d.3. A tank of water is covered with a 1 cm thick layer of linseed oil (noil = 1.48) above

which is air. What angle must a beam of light, originating in the tank, make at thewater-oil interface if no light is to escape?

4. Analyse the travel of a light ray through a parallel glass plate assuming refractiveindex of glass n. Determine displacement of the outgoing ray.

5. A point source S is located on the axis of, and 30 cm from a plane-convex thin lenswhich has a radius of 5 cm. The glass lens is immersed in air. Determine the locationof the image (a) when the flat surface is toward S and (b) when the curved surface istoward S.

31 Appendix 2A

R

d

A

B

Fig. 2.14 Propagation of light in U-tube.

6. Analyse multiple interference in a parallel plate FP etalon formed by two parallelplates with an active medium inside with gain g [3]. Derive an expression for a signalgain.

7. A glass rod of rectangular cross section is bent into U-shape as shown in Fig. 2.14[4]. A parallel beam of light falls perpendicularly on the flat surface A. Determinethe minimum value of the ratio R/d for which all light entering the glass throughsurface A will emerge from the glass through surface B. Assume that the refractiveindex of glass is 1.5.

8. The Sellmeier dispersion equation is an empirical expression for the refractive indexof a dielectric (like glass) in terms of the wavelength λ. One of the most popularmodels is [5], [6]

n2 = 1 + G1λ2

λ2 − λ21

+ G2λ2

λ2 − λ22

+ G3λ2

λ2 − λ23

(2.36)

Here G1, G2, G3 and λ1, λ2, λ3 are constants (called Sellmeier coefficients) whichare determined by fitting the above expression to the experimental data. Some of theSellmeier parameters are given in Table 2.2 (data combined from [5], [7] and [6]).

Table 2.2 Sellmeier parameters for SiO2.

Parameter G1 G2 G3 λ1 (µm) λ2 (µm) λ3 (µm)

SiO2 0.696749 0.408218 0.890815 0.0690660 0.115662 9.900559

Write a MATLAB program to plot refractive index as a function of λ from 0.5 µmto 1.8 µm for pure silica.

9. Using the Sellmeier relation, plot reciprocal group velocity and group velocity forSiO2. Assume 1.36 < λ[µm] < 1.65.

Appendix 2A: MATLAB listings

In Table 2.3 we provided a list of MATLAB files created for Chapter 2 and a short descriptionof each function.


Table 2.3 List of MATLAB functions for Chapter 2.

Listing Function name Description

2A.1 phase velocity.m Plots phase velocity2A.2 group velocity.m Plots group velocity2A.3 FP transmit.m Plots transmittance of Fabry-Perot etalon

Listing 2A.1 Program phase velocity.m. MATLAB program which plots phase velocity.

% File name: phase_velocity.m

% Illustrative plot of phase velocity

clear all

N_max = 101; % number of points for plot

t = linspace(0,30d-15,N_max); % creation of theta arguments

%

c = 3d14; % velocity of light in microns/s

n = 3.4; % refractive index

v_p = c/n; % phase velocity

lambda = 1.0; % microns

k = 2*pi/lambda;

frequency = v_p/lambda;

z = 0.6; % distance in microns

omega = 2*pi*frequency;

A = sin(k*z - omega*t);

plot(t,A,’LineWidth’,1.5)

xlabel(’Time’,’FontSize’,14);

ylabel(’Amplitude’,’FontSize’,14);

set(gca,’FontSize’,14); % size of tick marks on both axes

axis([0 30d-15 -1.5 2])

text(17d-15, 1.3, ’z(t)’,’Fontsize’,16)

line([1.53d-14,2d-14],[1,1],’LineWidth’,3.0) % drawing arrow

line([1.9d-14,2d-14],[1.1,1],’LineWidth’,3.0)

line([1.9d-14,2d-14],[0.9,1],’LineWidth’,3.0)

pause

close all

Listing 2A.2 Function group velocity.m. MATLAB program which plots group velocity.

% File name: group_velocity.m

% Illustrative plot of group velocity by superposition

% of two plane waves

clear all


33 Appendix 2A

t = linspace(0,300d-15,N_max); % creation of theta arguments

%

c = 3d14; % velocity of light in microns/s

n = 3.4; % refractive index

v_p = c/n; % phase velocity

lambda = 1.0; % microns

frequency = v_p/lambda;

z = 0.6; % distance in microns

omega = 2*pi*frequency;

k = 2*pi/lambda;

Delta_omega = omega/15.0;

Delta_k = k/15.0;

%

omega_1 = omega + Delta_omega;

omega_2 = omega - Delta_omega;

k_1 = k + Delta_k;

k_2 = k - Delta_k;

A = 2*cos(k*z - omega*t).*cos(Delta_k*z - Delta_omega*t);

plot(t,A,’LineWidth’,1.3)

xlabel(’Time’,’FontSize’,14)

ylabel(’Amplitude’,’FontSize’,14)


pause

close all

Listing 2A.3 Function FP transmit.m. Program plots transmittance of Fabry-Perot etalon.

% FP_transmit.m

% Plot of transmittance of FP etalon

clear all


t = linspace(0,1,N_max); % creation of theta arguments

delta = (5*pi)*t; % angles in radians

%

hold on

for r = [0.2 0.4 0.8] % reflection coefficient

F = 4*r^2/(1-r^2).^2; % coefficient of finesse

T = 1./(1 + F*(sin(delta/2)).^2);

plot(delta,T,’LineWidth’,1.5)

end

%

% Redefine figure properties

ylabel(’Transmittance’,’FontSize’,14)

xlabel(’Phase difference (rad)’,’FontSize’,14)



text(6.1, 0.05, ’2 \pi’,’Fontsize’,14)

text(12.2, 0.05, ’4 \pi’,’Fontsize’,14)

text(8.5, 0.1, ’r = 0.8’,’Fontsize’,14)

text(8.5, 0.5, ’r = 0.4’,’Fontsize’,14)

text(8.5, 0.8, ’r = 0.2’,’Fontsize’,14)

%

pause

close all

References

[1] F. L. Pedrotti and L. S. Pedrotti. Introduction to Optics. Third Edition. Prentice Hall,Upper Saddle River, NJ, 2007.

[2] E. Hecht. Optics. Fourth Edition. Addison-Wesley, San Francisco, 2002.[3] Y. Yamamoto. Characteristics of AlGaAs Fabry-Perot cavity type laser amplifiers. IEEE

J. Quantum Electron., 16:1047–52, 1980.[4] L. Yung-Kuo, ed. Problems and Solutions on Optics. World Scientific, Singapore, 1991.[5] J. W. Fleming. Dispersion in GeO2 − SiO2 glasses. Applied Optics, 23:4486–93, 1984.[6] S. O. Kasap. Optoelectronics and Photonics: Principles and Practices. Prentice Hall,

Upper Saddle River, NJ, 2001.[7] G. Ghosh, M. Endo, and T. Iwasaki. Temperature-dependent Sellmeier coefficients and

chromatic dispersions for some optical fiber glasses. J. Lightwave Technol., 12:1338–42, 1994.

3 Basic facts from electromagnetism

In this chapter we review foundations of electromagnetic theory and basic properties oflight.

3.1 Maxwell’s equations

Our discussion is based on Maxwell’s equations which in differential form are [1], [2]

∇ × E = −∂B

∂t(3.1)

∇ × H = J + ∂D

∂t(3.2)

∇ · D = ρv (3.3)

∇ · B = 0 (3.4)

whereE is electric field intensity [V/m]B is magnetic flux density [T ]H is magnetic field intensity [A/m]D is electric flux density [C/m2]J is electric current density [A/m2]ρv is volume charge density [C/m3].The operator ∇ in Cartesian coordinates is

∇ =[

∂

∂x,

∂

∂y,

∂

∂z

](3.5)

The above relations are supplemented with constitutive relations

D = εE (3.6)

B = μH (3.7)

J = σE (3.8)

where ε = ε0εr is the dielectric permittivity [F/m], μ = μ0μr is permeability [H/m], σ iselectric conductivity, εr is the relative dielectric constant. For optical problems consideredhere, μr = 1.

35


First, we recall two mathematical theorems, Gauss’s theorem∮S

F · ds =∫

V∇ · Fdv (3.9)

where S is the closed surface area defining volume V , and Stokes’s theorem∮L

F · dl =∫

A∇ × F · ds (3.10)

where contour L defines surface area A. With the help of the above theorems, Maxwell’sequations in differential form can be transformed into an integral form. Integral forms ofMaxwell’s equations are ∮

SD · ds =

∫V

ρvdv (3.11)∮S

B · ds = 0 (3.12)∮L

E · dl = −∫

A

∂B

∂t(3.13)∮

LH · dl =

∫A

∂D

∂t+ I (3.14)

Properties of the medium are mostly determined by ε, μ and σ . Further, for the dielectricmedium the main role is played by ε. The medium is known as linear if ε is independentof E; otherwise it is nonlinear. If it does not depend on position in space, the medium issaid to be homogeneous; otherwise it is inhomogeneous. If properties are independent ofdirection, the medium is isotropic; otherwise it is anisotropic.

3.2 Boundary conditions

Boundary conditions are derived from an integral form of Maxwell’s equations. For thatpurpose we separate all vectors into two components, one parallel to the interface and onenormal to the interface. The derivation of boundary conditions is facilitated by using thecontour and cylindrical shapes as shown in Fig. 3.1. It will be done independently for theelectric and magnetic fields.

3.2.1 Electric boundary conditions

First, analyse transversal components. Integrate Eq. (3.13) over closed loop C, see Fig. 3.1and then set �h → 0∫

ABCDA

E · dl = −E1 · dl + E2 · dl = −E1t�w + E2t�w = 0

37 Boundary conditions

A B

CD

Medium 2

Medium 1

n1

n2E2

D2

D1E1

εr1

εr2 Cw

hh

ρs

Fig. 3.1 An interface between two media. Contour and volume used to derive boundary conditions for fields between twodifferent dielectrics are shown.

Therefore

E1t = E2t (3.15)

The tangential component of the electric field is therefore continuous across the boundarybetween any two dielectric media. Using general relation (3.6) one obtains the boundarycondition for tangential component of vector D

D1t

εr1= D2t

εr2(3.16)

In order to obtain conditions for normal components, consider the cylinder shown in Fig. 3.1.Here, n1 and n2 are normal vectors pointing outwards of the top and bottom surfaces intocorresponding dielectrics. Apply Gauss’s law with integration over surface S of the cylinder∫

S

D · ds =∫

topD1 · n1ds +

∫bottom

D2 · n2ds = ρs�s

The contribution from the outer surface of the cylinder vanishes in the limit �h → 0. Usethe fact that n2 = −n1 and have

n1 · (D1 − D2) = ρs (3.17)

or

εr1E2n − εr2E1n = ρs (3.18)

when using general relation (3.6). Normal component of vector D is not continuous acrossboundary (unless ρs = 0).

3.2.2 Magnetic boundary conditions

When deriving boundary conditions for magnetic fields, we use approach similar as forelectric fields. We use Eq. (3.12) where integration is over the cylinder∫

S

B · ds = 0


Table 3.1 Summary of boundary conditions for electric and magnetic fields.

Field components General form Specific form

Tangential E n2 × (E1 − E2) = 0 E1t = E2 t

Normal D n2 · (D1 − D2) = ρs D1t − D2 t = ρs

Tangential H n2 × (H1 − H2) = Js H2 t = H1t

Normal B n2 · (B1 − B2) = 0 B1n = B2n

or

B1n = B2n (3.19)

Using general relation (3.7), one obtains condition for magnetic field H

μr1H1n = μr2H2n (3.20)

The above relations tell us that normal component of B is continuous across the boundary.To obtain how transversal magnetic components behave across interface, apply Ampere’s

law for contour C and then let �h → 0. One obtains∫C

H · dl =∫ B

AH2 · dl −

∫ D

CH1 · dl = I

Here I is the net current crossing the surface of the loop. As we let �h approach zero, thesurface of the loop approaches a thin line of length �w. Hence, the total current flowingthrough this thin line is I = Js ·�w, where Js is the magnitude of the normal component ofthe surface current density traversing the loop. We can therefore express the above equationas

H2t − H1t = Js (3.21)

Utilizing unit vector n2, the above relation can be written as

n2 × (H1 − H2) = Js (3.22)

where n2 is the normal vector pointing away from medium 2, see Fig. 3.1. Js is the surfacecurrent.

In Table 3.1 we summarize boundary conditions between two dielectrics for the electricand magnetic fields. The behaviour of various field components is shown schematically inFig. 3.2.

3.3 Wave equation

Here, we will derive the wave equation for a source-free medium where �v = 0 and J = 0.The wave equation is obtained by applying ∇× operation to both sides of Eq. (3.1). One

39 Time-harmonic fields

medium 1

medium 2

medium 1

medium 2

2nB

1nB

1tE 1tH

2tE 2tH

2nD

1nD

Fig. 3.2 Boundary conditions for fields.

obtains

∇ × ∇ × E = − ∂

∂t(∇ × B) = −μ

∂

∂t(∇ × H) = −μ

∂

∂t

∂D

∂t= −με

∂2D

∂t2

where we have used Maxwell Eq. (3.2) and relation (3.6). Next apply the following mathe-matical formula

∇ × ∇ × E = ∇(∇ · E) − ∇2E (3.23)

With the help of Maxwell Eq. (3.3), one finds finally

∇2E − με∂2

∂t2E (3.24)

which is the desired wave equation. The quantity με is related to velocity of light in avacuum as (assuming μr = 1)

με = μ0ε0εr = n2

c2(3.25)

where n is the refractive index of the medium.

3.4 Time-harmonic fields

In many practical situations fields have sinusoidal time dependence and are known astime-harmonic. This fact is expressed as

E(r, t) = Re{E (r) e jωt

}(3.26)

where E (r) is the phasor form of E (r, t) and is in general complex. Re {...} indicates‘taking the real part in’ quantity in brackets. Finally, ω is the angular frequency in [rad/s].In what follows, all fields will be represented in phasor notation.


Applying the time-harmonic assumption (3.26) to source-free Maxwell’s equations re-sults in

∇ × E = − jωμH (3.27)

∇ × H = jωεE (3.28)

Applying the time-harmonic assumption again to wave equation (3.24) gives

∇2E + k2E = 0 (3.29)

where k = ω√

με. Explicitly, the above wave equation is(∂2

∂x2+ ∂2

∂y2+ ∂2

∂z2+ k2

)Ei = 0 (3.30)

with i = x, y, z.

Example As an example, consider propagation of a uniform plane wave characterized by auniform electric field with nonzero component Ex. Assume also

∂2Ex

∂x2= 0,

∂2Ex

∂y2= 0 (3.31)

The wave equation reduces to

∂2Ex

∂z2+ k2Ex = 0 (3.32)

and has the following forward propagating solution:

Ex(z) = E0e jkz (3.33)

Magnetic field is determined from the Maxwell’s Equation (3.27)

∇ × E =∣∣∣∣∣∣

ax ay az

0 0 ∂∂z

Ex(z) 0 0

∣∣∣∣∣∣ = − jωμ(axHx + ayHy + azHz

)(3.34)

Here ax, ay, az are unit vectors along x, y, z axes, respectively. From the above equation, onefinds

Hx = 0 (3.35)

Hy = 1

jωμ

∂Ex(z)

∂z

Hz = 0

41 Time-harmonic fields

y

xH

E

H

Hk

k

k

E

E

z

Fig. 3.3 Visualization of the electric and magnetic fields.

Using the solution for Ex(z), one finally obtains the expression for magnetic field

H = ayHy(z) (3.36)

= 1

− jωμ(− jkEx)ay (3.37)

= ayk

ωμEx(z)

= ay1

ZEx(z)

where the impedance Z of the medium is defined as Z =√

μ

ε. The propagating wave is

shown in Fig. 3.3.We finish this section by providing a useful relation between electric and magnetic fields.

Assuming time-harmonic and plane-wave dependence of both fields as

E ∼ exp(iωt − ik · r)

and using constitutive relation (3.7), Maxwell Eq. (3.1) takes the form

k × E = ωμ0H

Introducing unit vector k along wave vector k and the expression for wave number k =ωn

√μ0ε0, one finds

n k × E = Z0H

where Z0 is the impedance of the free-space.


3.5 Polarized waves

Here we will discuss the concept of polarization of electromagnetic waves. Polarizationcharacterizes the curve which E vector makes (in the plane orthogonal to the directionof propagation) at a given point in space as a function of time. In the most general case,the curve produced is an ellipse and, accordingly, the wave is called elliptically polarized.Under certain conditions, the ellipse may be reduced to a circle or a segment of a straightline. In those cases it is said that the wave’s polarization is circular or linear, respectively.Since the magnetic field vector is related to the electric field vector, it does not need separatediscussion. First, consider a single electromagnetic wave.

3.5.1 Linearly polarized waves

Consider an electromagnetic wave characterized by electric field vector E directed alongthe x-axis

E = axE0 cos (ωt − kz + φ) (3.38)

It is known as a linearly polarized plane wave with the electric field vector oscillating inthe x direction. Wave propagates in the +z direction. In Eq. (3.38) ω = 2πν is the angularfrequency and k is the propagation constant defined as

k = ω

v

where v = c/n is the velocity of the electromagnetic wave in the medium having refractiveindex n. φ is known as a phase of electromagnetic wave.

The electromagnetic wave can also be written in the complex representation as

E = axE0ei(ωt−kz+φ) (3.39)

The actual field as described by Eq. (3.38) is obtained from (3.39) by taking real part. Amore general expression for the electromagnetic wave is

E = eE0ei(ωt−k·r+φ)

which is known as the plane polarized wave. Here unit vector e lies in the plane known asplane of polarization. It is perpendicular to vector k which describes direction of propagation

k · e = 0

3.5.2 Circularly and elliptically polarized waves

In general, when we have an arbitrary number of plane waves propagating in the samedirection they add up to a complicated wave. In the simplest case, one has only two suchplane waves. To be more specific, consider two plane waves oscillating along orthogonal

43 Polarized waves

directions. They are linearly polarized having the same frequencies and propagating in thesame direction

E1 = Exax = axE0x cos (ωt − kz) (3.40)

E2 = Eyay = ayE0y cos (ωt − kz + φ) (3.41)

We want to know the type of the resulting wave and the curve traced by the tip of the totalelectric vector E = E1 + E2

E = E1 + E2 = E0 {cos (ωt − kz) + cos (ωt − kz + φ)}First, eliminate cos (ωt − kz) term. From Eq. (3.40)

cos (ωt − kz) = Ex

Eox(3.42)

Use trigonometric identity

cos (α − β) = cos α cos β + sin α sin β

to express Eq. (3.41) as follows:

Ey = E0y

{cos (ωt − kz) cos δ + [

1 − cos2 (ωt − kz)]1/2

sin δ}

Substitute Eq. (3.42) in the above and have

Ey

E0y= Ex

E0xcos φ +

(1 − E2

x

E20x

)1/2

sin φ

Squaring both sides gives(Ey

E0y− Ex

E0xcos φ

)2

=(

1 − E2x

E20x

)sin2 φ

orE2

y

E20y

− 2 cos φEy

E0y

Ex

E0x+ E2

x

E20x

cos2 φ + E2x

E20x

sin2 φ = sin2 φ

Finally, the above equation gives(Ey

E0y

)2

+(

Ex

E0x

)2

− 2

(Ey

E0y

)(Ex

E0x

)cos φ = sin2 φ (3.43)

This is general equation of an ellipse. Thus the endpoint of E(z, t) will trace an ellipse at agiven point in space. It is said that the wave is elliptically polarized.

When phase φ = π2 , the resultant total electric field is(

Ey

E0y

)2

+(

Ex

E0x

)2

= 1

which describes right elliptically polarized wave since as time increases the end of elec-tric vector E rotates clockwise on the circumference of an ellipse. Typical situations areillustrated in Fig. 3.4 where we show elliptic, circular and linear polarizations.


zx

y

zx

y

zx

y

E

EE

(a) (b) (c)

Fig. 3.4 Typical states of polarization: (a) elliptic, (b) circular and (c) linear.

z

y

x

medium 1

medium 2

n

incident wave reflected wave

Fig. 3.5 Picture used in defining plane of incidence.

3.6 Fresnel coefficients and phases

In this section we discuss electromagnetic wave undergoing reflection at the boundarybetween two dielectrics, see Fig. 3.5. The plane of incidence is defined as the plane formedby unit vector n normal to the interface between two media and the directions of propagationof the incident and reflected waves.

We will derive so-called Fresnel coefficients [3] which are reflection and transmissioncoefficients expressed in terms of the angle of incidence and material properties (ε dielectricconstants) of the two dielectrics. Further, Fresnel phases are determined from Fresnelcoefficients using the following definition [1]:

r = e−2 jφ (3.44)

Both reflection coefficients r and phases φ will be calculated for both types of modes, TEand TM.

3.6.1 TE polarization

Referring to Fig. 3.6 the E1i, E1r, E2t are complex values of incident, reflected and transmit-ted electric fields in medium 1 and 2. The incident electric field in medium 1 (E1i) is parallelto the interface between both media. Such orientation is known as TE polarization. It is

45 Fresnel coefficients and phases

z

yx

E1i

θ1 θ1

θ2

H1i ki

krE1r

H1r

kt

E2t

H2t

medium 1

medium 2

Fig. 3.6 Fresnel reflection for TE polarization. Directions of vectors for TE polarized wave.

often said that electric field vector E is normal to the plane of incidence. Such configurationis also known as s-polarization.

Boundary conditions require that the tangential components of the total field E and thetotal field H on both sides of the interface be equal. Those conditions result in the followingequations:

E1i + E1r = E2t (3.45)

−H1i cos θ1 + H1r cos θ1 = −H2t cos θ2

We also have the following relations involving impedances in both media:

E1i

H1i= Z1,

E1r

H1r= Z1,

E2t

H2t= Z2 (3.46)

For a non-magnetic medium (μr = 1) the impedance can be written as Z = Z0n , with n

being the refractive index of the medium and

Z0 =√

μ0

ε0

is the impedance of a free space. Replacing magnetic fields in the second equation ofEq. (3.45) by the electric fields using Eq. (3.46) gives

E1i + E1r = E2t (3.47)

−E1i

Z1cos θ1 + E1r

Z1cos θ1 = −E2t

Z2cos θ2

Reflection coefficient is defined as

rT E = E1r

E1i(3.48)

From Eqs. (3.47) by eliminating E2t and using definition (3.48) one obtains 5

rT E = Z2 cos θ1 − Z1 cos θ2

Z2 cos θ1 + Z1 cos θ2


Replacing impedances Z by refractive indices n, we finally have

rT E = n1 cos θ1 − n2 cos θ2

n1 cos θ1 + n2 cos θ2

=n1 cos θ1 −

√n2

2 − n21 cos θ1

n1 cos θ1 +√

n22 − n2

1 cos θ1

(3.49)

using Snell’s law. For angles θ1 such that n22 − n2

1 cos θ1 < 0 the reflection coefficientbecomes complex. For such cases, we write it as

rT E =n1 cos θ1 − j

√n2

1 cos θ1 − n22

n1 cos θ1 + j√

n21 cos θ1 − n2

2

≡ a − jb

a + jb

Such complex number can be expressed as

rT E = e− jφT E

e jφT E= e−2 jφT E

where we have defined a + jb = e jφT E . Finally, using the definition of Fresnel phase,Eq. (3.44) one obtains

tan φT E =√

n21 sin2 θ1 − n2

2

n1 cos θ1(3.50)

The above equation represents phase shift during reflection for TE polarized electromagneticwave. For a more detailed discussion about phase change on reflection consult books byPedrotti [4] and Liu [5].

Example (a) Consider normal incidence of light on an air-silica interface. Compute thefraction of reflected and transmitted power. Also, express the transmitted loss in decibels.Assume refractive index of silica to be 1.45. (b) Repeat the calculations for Si which hasn = 3.50. (c) Consider coupling of GaAs optical source with a refractive index of 3.6 to asilica fibre which has refractive index of 1.48. Assume close physical contact of the fibreend and the source.

Solution(a) The corresponding coefficient known as reflectance [4] is

R = |r|2 =(

n1 − n2

n1 + n2

)2

(3.51)

Substituting values for air and silica, one has

R =(

1.45 − 1.00

1.45 + 1.00

)2

= 0.03

so about 3% of the light is reflected. The remainder, 97%, is transmitted.

47 Fresnel coefficients and phases

z

yx

medium 1medium 2

E1iH1i

ki

krE1r

H1r

kt

E2tH2t

θ1 θ1

θ2

Fig. 3.7 Fresnel reflection for TM polarization. Directions of vectors for a TM polarized wave.

The transmission loss in dB is

−10 log10 0.97 = 0.13[dB]

This result shows that there is about 0.2dB loss when light enters glass from air.(b) For Si, using the same procedure we obtain

R =(

1.00 − 3.50

1.00 + 3.50

)2

= 0.309

This means that about 31% of light is reflected.(c) The Fresnel reflection at the interface is

R =(

3.60 − 1.48

3.60 + 1.48

)2

= 0.174

Therefore about 17.4% of the created optical power is emitted back into the source. Thepower coupled into optical fibre is

Pcoupled = (1 − R) Pemitted

The power loss, α in decibels is

α = −10 log10

Pcoupled

Pemitted= −10 log10 (1 − R)

= −10 log10 ((1 − 0.174)) − 10 log10 (0.826) = 0.83[dB]

3.6.2 TM polarization

Field configuration used to analyse reflection for TM polarization (magnetic field parallelto the interface) is shown in Fig. 3.7. Here, electric field vector E is parallel to the plane ofincidence. Such configuration is also known as p-polarization.


We will derive coefficient of reflection rT M for TM mode. As before the E1i, E1r, E2t are(complex) values of incident, reflected and transmitted electric fields in medium 1 and 2.Similar notation holds for magnetic vectors. Boundary conditions require that tangentialcomponents are continuous across interface. The relevant conditions are

E1i cos θ1 − E1r cos θ1 = E2t cos θ2 (3.52)

H1i + H1r = H2t

Using Eqs. (3.46) for impedances to eliminate magnetic fields in the previous equationsresults in the following:

E1i cos θ1 − E1r cos θ1 = E2t cos θ2 (3.53)

E1i

Z1+ E1r

Z1= E2t

Z2

Reflection coefficient is defined as

rT M = E1r

E1i(3.54)

From Eqs. (3.53)

rT M = Z1 cos θ1 − Z2 cos θ2

Z1 cos θ1 + Z2 cos θ2

It can be also expressed in terms of refractive index

rT M = n2 cos θ1 − n1 cos θ2

n2 cos θ1 + n1 cos θ2

=n2 cos θ1 −

√n2

1 − n22 cos θ1

n2 cos θ1 +√

n21 − n2

2 cos θ1

(3.55)

TM phase is obtained in the same way as for TE polarization. Final result is

tan φT M = n21

n22

×√

n21 sin2 θ1 − n2

2

n1 cos θ1(3.56)

Again, for more information consult [4] and [5].

3.7 Polarization by reflection from dielectric surfaces

In interpreting the formulas for rT E and rT M we distinguish between two situations:

(1) for n1 <n2 or n = n2n1

> 1, one defines so-called external reflection,

(2) for n1 >n2 or n = n2n1

< 1, one defines so-called internal reflection.

Example of (1) is air-to-glass reflection and example of (2) is the glass-to-air reflection.A plot of coefficients of reflection for n = 1.50 is shown in Fig. 3.8. MATLAB code is

49 Polarization by reflection from dielectric surfaces

0 20 40 60 80 100−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

Angle of incidence

rTE

rTM

θB

Fig. 3.8 Reflections of TE and TMmodes for external reflection with n = 1.50. Brewster’s angle is also shown.

medium 1 medium 2

polarized

partiallypolarized

unpolarized Bθ

Fig. 3.9 Polarization at Brewster’s angle.

shown in Appendix, Listing 3A.1. The so-called Brewster’s angle for which rT M = 0 isalso shown. For more information, see the book by Pedrotti and Pedrotti [4].

One can obtain polarization by reflection at Brewster’s angle, see Fig. 3.9. In the figurewe illustrate properties of the reflected and transmitted light incident at the surface atBrewster’s angle.

Unpolarized light is incident at the interface at Brewster’s angle θB. Upon reflection oneobtains polarized TM light. At the same time, the refracted light is only partially polarizedsince rT E is nonzero.

At Brewster’s angle θB of incidence, coefficient of reflection rT M (also known as r‖ sinceE is parallel to the plane of incidence) is zero; i.e. r‖ = 0. The above happens when the


sum of the angles of incidence and refraction is equal to π/2

θ1 + θ2 = π

2

No such angle exists for the s polarization (TE). Thus if an unpolarized light is incident atBrewster’s angle, the reflected light will be linearly polarized, in fact s-polarized.

3.7.1 Expression for Brewster’s angle

Expression for rT M is

rT M =n2 cos θ1 − n1

n2

√n2

2 − n21 sin2 θ1

n2 cos θ1 + n1n2

√n2

2 − n21 sin2 θ1

Vanishing of rT M corresponds to θ1 = θB. We have

n2 cos θB − n1

n2

√n2

2 − n21 sin2 θB = 0

from which we find an expression for Brewster’s angle

tan θB = n2

n1

3.8 Antireflection coating

In Chapter 9 on semiconductor optical amplifiers (SOA) we will discuss why there is aneed to reduce (or completely eliminate) the effect of reflections. Therefore, the practicalquestion is how to eliminate reflections at the interface between two dielectrics.

As seen before, light passing through a boundary between two dielectrics is lost due toreflection, see Eq. (3.51). The effect depends on the difference of the values of refractiveindices between neighbouring layers. One can observe that for equal refractive indices therewill be no reflection but also no refraction, which is not a very interesting possibility.

A practical method of reducing reflections is to use several layers with properly selectedvalues of refractive indices. For a single layer, interference of two reflected waves can leadto elimination of reflection, but only at a particular wavelength.

Let us analyse this situation in more detail. Consider reflections of two waves R1 and R2,see Fig. 3.10. One of the conditions for destructive interference is that the amplitudes ofboth reflected waves should be equal. From Eq. (3.51), reflection of ray R1 at interface ‘a’is described by reflection coefficient

R =(

n0 − n f

n0 + n f

)2

51 Antireflection coating

h

I

T

R1

R 2

nf

ns

n0a

b

Fig. 3.10 Illustration of reflection from single dielectric film layer. Case of destructive interference is shown.

whereas reflection of ray R2 at interface ‘b’ is described by coefficient

R =(

ns − n f

ns + n f

)2

Both coefficients should be equal which gives(n0 − n f

n0 + n f

)2

=(

ns − n f

ns + n f

)2

Assuming that the upper medium is air, i.e. n0 = 1, from the above one finds expressionfor the value of refractive index for a layer as

n f =√

ns (3.57)

Therefore, at a particular wavelength there will be no reflection once the refractive indexof coating layer is given by (3.57). If there is a need to design structures producing noreflection over some frequency band, one must design multilayer structure consisting oflayers with different refractive indices. In order to design such structures a more realisticdescription is necessary. Typical calculations are based on transfer matrix approach.

3.8.1 Transfer matrix approach

We now describe the transfer matrix approach, also known as the characteristic matrixapproach [6].

Consider a single dielectric layer with refractive index n deposited on a substrate withrefractive index ns. Above it is a medium with refractive index n0, see Fig. 3.11.

We consider an s-polarized wave where electric field vector E is perpendicular to theplane of incidence. Boundary conditions dictate that tangential components of E and Hfields are continuous across the interfaces, i.e. at positions z = z1 and z = z2.

We introduce the convention that E+(z) denotes light propagating in the positive zdirection whereas E−(z) denotes light propagating in the negative z direction. Close tointerface described by z = z1, due to continuity of tangential component one writes

E(z1) = E+(z+1 ) + E−(z+

1 ) = E+(z−1 ) + E−(z−

1 ) (3.58)


n

z

z1

z2

n0

α0

αα

αsns

E ( )- z1-

E ( )z2

E ( )+ +

+ ++

z1

E ( )z2-

E ( )- z2-

E

Fig. 3.11 Illustration of electromagnetic fields at the boundaries used in the derivation of transfer matrix.

where z+1 signifies value of coordinate z slightly larger than z1, etc. Similarly, close to z = z2

E(z2) = E+(z+2 ) + E−(z+

2 ) = E+(z−2 ) + E−(z−

2 ) (3.59)

For future use we introduce d = z2 − z1. When field E(z) propagates in the film overdistance d, it acquires phase δ derived previously (see Eq. 2.31)

δ = 2π

λ0nd cos α (3.60)

Electric fields that are close to interfaces are therefore related as

E+(z−2 ) = E+(z+

1 )e−iδ (3.61)

and

E−(z−2 ) = E−(z+

1 )eiδ (3.62)

To determine the behaviour of a magnetic field, we use the fact that fields E and H arerelated as

H = n

Z0k × E

where k is the unit vector along the wave vector k and Z0 =√

μ0

ε0is the impedance of the

free space. The condition of continuity of tangential component of H vector gives:at z = z1

Z0H(z1) = [E+(z−

1 ) − E−(z−1 )]

n0 cos α0 = [E+(z+

1 ) − E−(z+1 )]

n cos α (3.63)

at z = z2

Z0H(z2) = [E+(z−

2 ) − E−(z−2 )]

n cos α = E+(z+2 )ns cos α (3.64)

Substitute Eqs. (3.61) and (3.62) involving phase δ into (3.59) and (3.64) to obtain

E(z2) = E+(z+1 )e−iδ + E−(z+

1 )eiδ

H(z2) = n cos α

Z0

[E+(z+

1 )e−iδ − E−(z+1 )eiδ

]

53 Bragg mirrors

Solutions of the above equations are

2E−(z+1 )eiδ = E(z2) − Z0

n cos αH(z2)

2E+(z+1 )e−iδ = E(z2) + Z0

n cos αH(z2)

With the help of the above solutions and using Eqs. (3.58) and (3.63) we can evaluate fieldsat z1 and z2

E(z1) = E(z2) cos δ + Z0

n cos αH(z2)i sin δ

and

Z0H(z1) = E(z2)i sin δn cos α + Z0H(z2) cos δ

The above equations can be written in matrix form[E(z1)

Z0H(z1)

]=[

cos δ i sin δn cos α

in sin δ cos α cos δ

] [E(z2)

Z0H(z2)

]= ←→

M

[E(z2)

Z0H(z2)

](3.65)

For normal incidence, i.e. α = 0, the transfer matrix is

←→M =

[cos δ i sin δ

nin sin δ cos δ

](3.66)

Special cases of importance are:1. The quarter-wave layer when nd = λ0/4; δ = π/2 and matrix

←→M is

←→M λ/4 =

[0 i

n

in 0

](3.67)

2. The half-wave layer when nd = λ0/2; δ = π and matrix←→M is

←→M λ/2 =

[−1 00 −1

](3.68)

3.9 Braggmirrors

The Bragg mirror, also known as the Bragg reflector, consists of identical layers of dielectricswith high and low values of refractive indices as shown in Fig. 3.12. The main interest infabricating such structures is that they have extremely high reflectivities at optical andinfrared frequencies. They are important elements of VCSELs where high reflectivity andbandwidth are required. A typical structure forming Bragg mirror consists of N layers ofdielectrics with refractive indices nL (low refractive index) and nH (high refractive index).The ratio of those values, the so-called contrast ratio, plays an important role.

The structure is known as a quarter-wave dielectric stack, which means that the opticalthicknesses are quarter-wavelength long; that is nH ·aH = nL·aL = λ0/4 at some wavelength


yE

z

y xaHaL

Hn Hn Hn HnLn Ln Ln

Fig. 3.12 Schematic of a seven-layer dielectric (Bragg) mirror; shown areN = 3 periods.

xn-2 xn-1 xn

(n-1) th cell nth cell

xp

aH aHaL aL

nL nLnH nH

Cn–1Dn–1

An–1Bn–1

Cn AnBnDn

Fig. 3.13 Notation used in the analysis of Bragg mirrors.

λ0. The structure consists of an odd number of layers with the high index layer being thefirst and the last layers [7].

Here, we outline a formalism which is used in the design of Bragg mirrors. We alsoillustrate its applications with an example. We restrict our discussion to s-polarization (TEmodes). Electric field vector is directed along y-axis E = E(x) · ay, see Fig. 3.12 where weshowed the orientation of electric fields and analysed structure.

We develop transfer matrix notation which is based on the transfer of fields from cell nto cell n − 1. The cell is formed by two neighbouring layers which create a basic periodicstructure which is then repeated some number of times. The electric field will now beexpressed in the three layers of interest, see Fig. 3.13 for notation.

In cell n one expresses the electric field as

Ey(x) ={

An e− jkH (x−xn ) + Bn ejkH (x−xn ), xp < x < xn

Cn e− jkL(x−xp) + Dn e jkL(x−xp), xn < x < xp

(3.69)

where k2H = (

nH ωc

)2and k2

L = (nLω

c

)2. One has also the following relations for coordinates

of layers: xn = n · �, xn−1 = (n − 1) · �, xp = n · � − aH . Here � = aH + aL is the periodof the structure.

55 Bragg mirrors

The magnetic field vector is determined from the Maxwell equation, Eq. (3.1). For ourgeometry, one has the relation between electric and magnetic fields

∂Ey(x)

∂x= − jωμ0Hz (3.70)

From continuity of electric field at xp and xn−1 one obtains

x = xp Cn + Dn = An ejkH ·aH + Bn e− jkH ·aH

x = xn−1 An−1 + Bn−1 = Cn e jkL·aL + Dn e− jkL·aL(3.71)

Similarly, continuity of magnetic field produces equations

x = xp −kLCn + kLDn = −kH An e jkH ·aH + kH Bn e− jkH ·aH

x = xn−1 −kH An−1 + kH Bn−1 = −kLCn e jkL·aL + kLDn e− jkL·aL(3.72)

The previous four equations can be written in matrix forms as follows:[1 1kL −kL

] [Cn

Dn

]=[

e− jkH aH e jkH aH

kH e− jkH aH −kH ejkH aH

] [An

Bn

]and [

1 1kH −kH

] [An−1

Bn−1

]=[

e jkLaL e− jkLaL

kLe jkLaL −kLe− jkLaL

] [An

Bn

]The above equations transfer fields from cell n to cell n − 1. In the next step amplitudes[

Cn

Dn

]will be eliminated and the transfer process will be described in terms of amplitudes[

An

Bn

]. By doing so, one obtains the practical equation

[An−1

Bn−1

]= ←→

T

[An

Bn

](3.73)

Matrix←→T can be expressed as

←→T = ←→

X H · ←→T L · ←→

X L · ←→T H (3.74)

Matrix←→T L describes propagation in a uniform medium having refractive index nL and

thickness aL and has the form

←→T L =

[e jkLaL 0

0 e− jkLaL

](3.75)

Similarly, matrix←→T H describes propagation in a uniform medium having refractive index

nH and thickness aH . Matrices←→X H and

←→X L describe behaviour at interfaces and are

←→X H =

⎡⎢⎢⎣kH + kL

2kH

kH − kL

2kH

kH − kL

2kH

kH + kL

2kH

⎤⎥⎥⎦ (3.76)


Hn

2th cell Nth cell1th cell

Hn Hn HnLn Ln Ln

A0

B0 AN

Fig. 3.14 Notation used in the definition of reflection from Bragg mirror.

and

←→X L =

⎡⎢⎣kL + kH

2kL

kL − kH

2kLkL − kH

2kL

kL + kH

2kL

⎤⎥⎦ (3.77)

By referring to the Fig. 3.13 propagation (to the left) from an arbitrary point in the struc-ture, say point xn involves matrix

←→T H (propagation over a uniform region with nH , then

propagation over an interface from region having nH to nL (described by matrix←→X L), prop-

agation over uniform region with nL (described by matrix←→T L) and finally, propagation

over an interface from nL to nH (described by matrix←→X H ).

Matrix←→T can be expressed as

←→T =

[a bc d

](3.78)

where the elements are

a = eiaH kH

[cos aLkL + i

2

(kH

kL+ kL

kH

)sin aLkL

](3.79)

b = i

2e−iaH kH

(kL

kH− kH

kL

)sin aLkL (3.80)

c = i

2eiaH kH

(kH

kL− kL

kH

)sin aLkL (3.81)

d = e−iaH kH

[cos aLkL − i

2

(kH

kL+ kL

kH

)sin aLkL

](3.82)

Matrix←→T is unimodular matrix, i.e. real square matrix with determinant ad − bc = ±1

[8].Referring to Fig. 3.14, the amplitude reflection coefficient is defined as

r = B0

A0(3.83)

Choosing a column vector of layer one as the zeroth unit cell and remembering that all cellsare identical, one obtains for N cells[

A0

B0

]=[

a bc d

]N [An

Bn

]= ←→

T N

[An

Bn

]

57 Bragg mirrors

1000 1200 1400 1600 1800 2000 22000

0.2

0.4

0.6

0.8

1

1.2

Wavelenght (nm)

Ref

lect

ivity

Fig. 3.15 Spectrum of reflectivity of a Bragg mirror.

Since matrix←→T is a unimodular matrix, one has the following relation [8][a bc d

]N

=[

a · UN−1(x) − UN−2(x) b · UN−1(x)

c · UN−1(x) d · UN−1(x) − UN−2(x)

]

where x = 12 (a + d) and UN (x) are Chebyshev polynomials of the second kind

Um(x) = sin[(m + 1) cos−1 x

]√

1 − x2= sin [(m + 1) θ ]

sin θ

where x = cos θ . Using the above relations, the amplitude reflection coefficient is evaluatedas [

A0

B0

]=[

a · UN−1(x) − UN−2(x) b · UN−1(x)

c · UN−1(x) d · UN−1(x) − UN−2(x)

][AN

0

]for cell N . From above relation, one finds

r = B0

A0= c · UN−1(x)

a · UN−1(x) − UN−2(x)

The definition of reflectance is [9] R = |r|2. After a little algebra one finds

R = |c|2|c|2 + (

sin K�sin NK�

)2(3.84)

Spectrum of reflectivity as given by Eq. (3.84) for N = 10 periods of Bragg reflector isshown in Fig. 3.15. The MATLAB code is shown in Appendix, Listing 3A.2. The followingparameters were adopted: nH = 2.25, nL = 1.45, aH = 167 nm and aL = 259 nm.


A B

x

z

sxn 2

zs2

θ

virtual reflecting plane

reflecting interface0n 1 n 2>

Fig. 3.16 The lateral displacement of a light beam on total reflection at the interface between two dielectric media(Goos-Hanchen shift).

3.10 Goos-Hanchen shift

Detailed examination of the internal reflection of a light beam on a planar dielectric interfaceshows that the reflected beam is shifted laterally from the trajectory predicted by the simpleray analysis, as shown in Fig. 3.16. The lateral displacement arises because wave energyactually penetrates beyond the interface into the lower index media before turning around.Total lateral shift is d = 2zs whereas xs is penetration depth.

To determine lateral shift d, we follow Kogelnik [10]. Consider incident beam A asconsisting of two plane waves incident at two slightly different angles with wave vec-tors β ± �β. The complex amplitude A(z) of the incident wave packet at the interfacex = 0 is

A(z) = e− j(β+�β)z + e− j(β−�β)z

= e− jβz(e− j�βz + e j�βz

)= 2 cos (�βz) e− jβz (3.85)

Reflected beam B is

B = R · A (3.86)

where R = exp (2 jφ). Phase shift φ that occurs upon reflection is different for both typesof polarization and has been determined before, see Eqs. (3.50) and (3.56). Phase shift φ isa function of angle θ and propagation constant β. Assume a small value of �β and expand

φ (β + �β) = φ (β) + dφ

dβ�β ≡ φ (β) + �φ

Using the above expansion and (3.85) allows us to write reflected beam B as

B = R · A = e2 jφ(β±�β){e− j(β+�β)z + e− j(β−�β)z

}= e− j(βz−2φ)

{e j(�βz−2�φ) + e− j(�βz−2�φ)

}= 2e− j(βz−2φ) cos (�β − 2zs) (3.87)

59 Poynting theorem

where zs = dφ

dz . The net effect of the phase shift is to displace the beam along the z axis bya distance 2zs. Distance zs is different for both types of modes. For TE modes [11]

k0zs = tan θ(β2 − n2

c

)1/2

whereas for TM modes

k0zs = tan θ(β2 − n2

c

)1/2

1

β2/n2f + β2/n2

c − 1

3.11 Poynting theorem

Electromagnetic waves carry with them electromagnetic power. We will now deliver a rela-tion between the rate of such energy transfer and the electric and magnetic field intensities.We start with Maxwell’s equations (3.1) and (3.2)

∇ × E = −∂B

∂t(3.88)

∇ × H = ∂D

∂t+ J (3.89)

Use the following mathematical identity:

∇ · (E × H) = H · (∇ × E) − E(∇ × H) (3.90)

Substituting Maxwell’s equations into (3.90) and using constitutive relations, one obtains

∇ · (E × H) = −H · μ∂H

∂t− E · J − E · μ

∂E

∂t

= −μ1

2

∂H · H

∂t− σE2 − ε

1

2

∂E · E

∂t

= − ∂

∂t

(1

2εE2 + 1

2μH 2

)− σE2 (3.91)

Integrate (3.91) over volume V and apply Gauss’s theorem, and then one finds∫V

∇ · (E × H)dv = − ∂

∂t

∫V

(1

2εE2 + 1

2μH 2

)dv −

∫V

σE2dv (3.92)

The Poynting vector P is defined as

P = E × H (3.93)

Eq. (3.92) can be written as

−∮

SP · ds = ∂

∂t

∫V

(we + wm) dv +∫

Vpσ dv (3.94)


where

we = 12εE2 is electric energy density

wm = 12μH 2 is magnetic energy density

pσ = σE2 is Ohmic power density

From the above equation, one can interpret vector P as representing the power flow perunit area.

3.12 Problems

1. Using the expression derived for a transfer matrix, consider an antireflection structureconsisting of two-layer quarter-wave-thickness films made of CeF3 (low index layerwith refractive index n1 = 1.65) and high-index layer of ZrO2 (refractive indexn = 2.1) deposited on glass substrate with refractive index ns = 1.52. Assume thaton top there is air with n0 = 1. Plot the spectral reflectance of such structure.

2. Show that a linearly polarized plane wave can be decomposed into a right-hand anda left-hand circularly polarized waves of equal amplitudes.

3. Derive an expression for a transfer matrix assuming p-polarization. In this case thevector of electric field E is parallel to the plane of incidence.

4. Determine the reflectance for normal incidence and plot it versus path differ-ence/wavelength for various values of refractive index.

5. Write a MATLAB program to illustrate propagation of EM wave elliptically po-larized. The wave will consist of two perpendicular waves of unequal amplitudes.Visualize propagation in 3D. Analyse the state of polarization (SOP) by changingphase difference φ between individual waves.

6. Analyse a Bragg mirror for TM polarization. Derive matrix elements of matrix←→T

for this polarization [9].7. Assume that for some materials refractive index at a particular wavelength is negative.

Discuss the consequences of such an assumption. Consider modification of Snell’slaw.

8. What percentage of the incoming irradiance is reflected at an air-glass (nglass = 1.5)interface for a beam of natural light incident at 70◦?

9. Write an expression for a right circularly polarized wave propagation in the positivez-direction such that its E-field points in the negative x-direction at z = 0 and t = 0.

10. Verify that linear light is a special case of elliptical light.

3.13 Project

1. Write a MATLAB program to determine reflectance for a multilayered structure con-sisting of different dielectric materials of various thicknesses. Analyse the special caseof d = λ/2.

61 Appendix 3A



3A.1 reflections TE TM .m Plots reflections based on Fresnel equations for TE andTM modes

3A.2 Bragg an.m Plots reflectivity spectrum of Bragg mirror for TE modeusing an analytical method

2. Develop MATLAB functions to analyse propagation of TM modes for arbitrary mul-tilayered waveguiding structure. Use a similar approach as described above for TEmodes.


In Table 3.2 we provide a list of MATLAB files created for Chapter 3 and a short descriptionof each function.

Listing 3A.1 Program reflections TE TM.m. Program plots reflection coefficients for TEand TM modes.

% reflections_TE_TM.m

% Plot of reflections based on Fresnel equations

% TE and TM reflections

% cases of external reflection

clear all

n = 1.50; % relative refractive index


t = linspace(0,1,N_max); % creation of theta arguments

theta = (pi/2)*t; % angles in radians

% Plot reflection coefficients

num_TE = cos(theta) - sqrt(n^2-(sin(theta)).^2 );

den_TE = cos(theta) + sqrt(n^2-(sin(theta)).^2 );

r_TE = num_TE./den_TE; % Plot for TE mode

num_TM = n^2*cos(theta) - sqrt(n^2-(sin(theta)).^2 );

den_TM = n^2*cos(theta) + sqrt(n^2-(sin(theta)).^2 );

r_TM = num_TM./den_TM; % Plot for TM mode

%

angle_degrees = (theta./pi)*180;

x_line = [0 max(angle_degrees)]; % needed to draw horizontal line

y_line = [0 0]; % passing through zero

plot(angle_degrees,r_TE,angle_degrees,r_TM, x_line, y_line, ’-’,...

’LineWidth’,1.5)


xlabel(’Angle of incidence’,’FontSize’,14)


text(60, -0.6, ’r_{TE}’,’Fontsize’,14)

text(60, -0.2, ’r_{TM}’,’Fontsize’,14)

text(55, 0.1, ’\theta_{Brewster}’,’Fontsize’,14)

pause

close all

Listing 3A.2 Program Bragg an.m. Determines spectrum of reflectivity of a Bragg mirrorfor TE mode using an analytical method.

% File name: bragg_an.m

% Determines reflectivity spectrum of Bragg mirror for TE mode

% using analytical method

clear all

N = 10; % number of periods

n_L = 1.45; % refractive index

n_H = 2.25; % refractive index

a_L = 259; % thickness (nm)

a_H = 167; % thickness (nm)

Lambda = a_L + a_H; % period of the structure

lambda = 1000:10:2200;

k_L = 2*pi*n_L./lambda;

k_H = 2*pi*n_H./lambda;

%

a=exp(1i*a_H*k_H).*(cos(k_L*a_L)+(1i/2)*(k_H./k_L+k_L./k_H).*sin(k_L*a_L));

d=exp(-1i*a_H*k_H).*(cos(k_L*a_L)-(1i/2)*(k_H./k_L+k_L./k_H).*sin(k_L*a_L));

b = exp(-1i*a_H*k_H).*((1i/2)*(k_L./k_H - k_H./k_L).*sin(k_L*a_L));

c = exp(1i*a_H*k_H).*((1i/2)*(k_H./k_L - k_L./k_H).*sin(k_L*a_L));

%

K = (1/Lambda)*acos((a+d)/2);

tt = (sin(K*Lambda)./sin(N*K*Lambda)).^2;

denom = abs(c).^2 + tt;

R = abs(c).^2./denom;

plot(lambda,R,’LineWidth’,1.5)

axis([1000 2200 0 1.2])

xlabel(’Wavelenght (nm)’,’FontSize’,14)

ylabel(’Reflectivity’,’FontSize’,14)


pause

close all

63 References

References

[1] J. D. Kraus. Electromagnetics, Fourth Edition. McGraw-Hill, New York, 1992.[2] D. K. Cheng. Field and Wave Electromagnetics. Second Edition. Addison-Wesley

Publishing Company, Reading, MA, 1989.[3] Z. Popovic and B. D. Popovic. Introductory Electromagnetics. Prentice-Hall, Upper

Saddle River, NJ, 2000.[4] F. L. Pedrotti and L. S. Pedrotti. Introduction to Optics. Third Edition. Prentice Hall,

Upper Saddle River, NJ, 2007.[5] J.-M. Liu. Photonic Devices. Cambridge University Press, Cambridge, 2005.[6] A. Thelen. Design of Optical Interference Coatings. McGraw-Hill, New York, 1989.[7] R. S. Geels. Vertical-Cavity Surface-Emitting Lasers: Design, Fabrication and Char-

acterization. PhD thesis, University of California, Santa Barbara, 1991.[8] E. W. Weisstein, Unimodular Matrix. From MathWorld–A Wolfram Web Resource,

http://mathworld.wolfram.com/UnimodularMatrix.html, 3 July 2012.[9] P. Yeh, A. Yariv, and C.-S. Hong. Electromagnetic propagation in periodic stratified

media. I. general theory. J. Opt. Soc. Am., 67:423–38, 1977.[10] H. Kogelnik. Theory of optical waveguides. In T. Tamir, ed., Springer Series in Elec-

tronics and Photonics. Guided-Wave Optoelectronics, volume 26, pp. 7–88. Springer-Verlag, Berlin, 1988.

[11] C. R. Pollock and M. Lipson. Integrated Photonics. Kluwer Academic Publishers,Boston, 2003.

4 Slab waveguides

Slab waveguides are important ingredients of both passive and active devices. Therefore,discussion and understanding of their properties are critical. The following topics will bediscussed in this chapter:

• ray optics of the slab waveguide• electromagnetic description• three-layer problem for symmetric guiding structure• TE modes in multilayer waveguide.

4.1 Ray optics of the slab waveguide

In this section we discuss applications of ray optics concepts to analyse the propagationof light in slab waveguides. We start with summarizing the concept of numerical aperture(NA).

4.1.1 Numerical aperture

Consider light entering a waveguide from air, having the refractive index n0 to waveguidingstructure with refractive indices n1 and n2 (see Fig. 4.1). We represent light within ray optics.We want to understand the conditions under which light will propagate in the middle layerwith the value of refractive index n1.

Ray 1 (dashed line) which enters waveguide at large incident angle θ1 propagates throughthe middle layer and penetrates the upper layer with refractive index n2. Since this ray doesnot propagate in the middle layer, it is effectively lost.

As we gradually decrease incident angle θ , we reach the situation where a ray slides acrossthe interface (case for ray 3). In such a case incident angle θ is known as the acceptanceangle θa. The internal angle of incidence at point D is then φc and it is determined from therelation

sin φc = n2

n1(4.1)

For the incident ray (like ray 2) which enters the interface at an angle smaller than theacceptance angle θ < θa, it will propagate in the middle layer. Let us analyse the existingsituation in more detail.

64

65 Ray optics of the slab waveguide

1

2

a

n1

n2

n2

A Cφφ

D

1

1

2

2

3

3

B

c

n0

θ

θ

θ

Fig. 4.1 Light entering slab waveguide.

From triangle ABC, one finds θ2 = π/2 − φ (angle θ2 not shown in Fig. 4.1). FromSnell’s law for ray 3, we obtain

n0 sin θ1 = n1 sin θ2 = n1 sin (π/2 − φ) = n1 cos φ = n1(1 − sin2 φ

)1/2

From the above, we have (using Eq. 4.1) for ray 2 (propagating at the acceptance angle, i.e.for θ1 = θa)

n0 sin θa = n1(1 − sin2 φc

)1/2 = (n2

1 − n22

)1/2

Numerical aperture (NA) is defined as

NA ≡ n0 sin θa = (n2

1 − n22

)1/2(4.2)

We introduce relative refractive index difference � as

� ≡ n21 − n2

2

2n21

The approximate expression for � is obtained by observing that n1 ≈ n2, which allows usto write n1 + n2 ≈ 2n1. The approximate formula for � is therefore

� = (n1 + n2) (n1 − n2)

2n21

≈ 2n1 (n1 − n2)

2n21

= n1 − n2

n1

In terms of �, NA can be approximately expressed as

NA = n1 (2�)1/2

4.1.2 Guidedmodes

We will first discuss the propagation of a light ray in a waveguide formed by the film ofdielectric surrounded from below by substrate and by cover layer from above, see Fig. 4.2. Insuch a structure light travels in a zig-zag fashion through the film. Light is monochromaticand coherent and is characterized by ω angular frequency and λ free-space wavelength.The following relation holds k = 2π

λ, where k is known as the wavenumber. Electric field

of the propagating wave is

E ∼ e− jkn f (±x cos φ+z sin φ)

66 Slab waveguides

filmnfh

x

φ

covernc

substratens

z

Fig. 4.2 Light propagation in a slab waveguide in a ray approximation.

Let β be a propagation constant of the guided mode of the slab. One has

β = ω

vp= kn f sin φ (4.3)

where vp is the phase velocity. Please note that not all angles are allowed (sometimes none).Referring to Fig. 4.2, rays at φ = 90◦ travel horizontally along the waveguide. Generally,rays which undergo total internal reflection can travel within the waveguide. That conditioncan be fulfilled for rays travelling at angles fulfilling the condition (all angles are determinedwith respect to normal)

φc ≤ φ ≤ 90◦

or

n f sinφc ≤ n f sinφ ≤ n f sin90◦ = n f

where we have also multiplied the inequality by n f . Using Eqs. (4.1) and (4.3), one obtains

ns ≤ N ≤ n f (4.4)

Here we introduced ‘effective guide index’ N defined as

N = β

k= n f sinφ (4.5)

4.1.3 Transverse resonance condition

The condition is expressed as ‘the sum of all phase shifts during propagation must be amultiple of 2π ’. The following are the contributions to the phase change:

kn f h cos φ phase change during first transverse passage through the film−2φc phase shift on total reflection from cover−2φs phase shift on total reflection from substrate

Combining the above contributions, one obtains transverse resonance condition as

2kn f h cos φ − 2φs − 2φc = 2πν, ν= 0, ± 1, ± 2, . . . (4.6)

which is the dispersion equation of the guide. Variable ν identifies mode number.

67 Ray optics of the slab waveguide

Expression for the Fresnel phase for TE polarization was derived in Chapter 3, Eq. (3.50).Modifying notation slightly (angle θ1 is replaced by φ), the relation reads

tan φ =√

n21 sin2 φ − n2

2

n1 cos φ

Adopting the above expression for our present geometry, one writes for phases on reflec-tions:

1. for reflection on the film-cover interface

tan φc =√

n2f sin2 φ − n2

c

n f cos φ=√

n2f sin2 φ − n2

c√n2

f − n2f sin2 φ

(4.7)

2. for reflection on the film-substrate interface

tan φs =√

n2f sin2 φ − n2

s

n f cos φ=√

n2f sin2 φ − n2

s√n2

f − n2f sin2 φ

(4.8)

4.1.4 Transverse condition: normalized form

The above transverse condition will now be cast in a normalized form which is more suitablefor numerical work. For that, let us introduce the following variables: variable V

V ≡ k · h√

n2f − n2

s (4.9)

normalized guide index, b

b ≡ N2 − n2s

n2f − n2

s

(4.10)

and asymmetry parameter for TE modes, a

a ≡ n2s − n2

c

n2f − n2

s

(4.11)

For TE modes, use transverse resonance condition and Fresnel phases, and obtain

V√

1 − b = ν · π + tan−1

√b

1 − b+ tan−1

√b + a

1 − b(4.12)

Example Derive transverse resonance condition in a normalized form, Eq. (4.12).

SolutionUsing definitions (4.10) and (4.11), one can prove the following relations:

b

1 − b= N2 − n2

s

n2f − N2

68 Slab waveguides

and

b + a

1 − b= N2 − n2

c

n2f − N2

and also

1 − b = n2f − N2

n2f − n2

s

Using the above relations, the expressions for phases, Eqs. (4.7) and (4.8) can be written as

φc = tan−1

√√√√N2 − n2c

n2f − N2

= tan−1

√b + a

1 − b

and

φs = tan−1

√√√√N2 − n2s

n2f − N2

= tan−1

√b

1 − b

One also has

n f cos φ =√

n2f − n2

f sin2φ =

√n2

f − N2

Substituting the above formulas into (4.6), we obtain

V√

1 − b = ν · π + tan−1

√b

1 − b+ tan−1

√b + a

1 − b

A MATLAB program to analyse the transverse resonance condition (4.12) which plotsnormalized guide index b versus variable V , is provided in the Appendix, Listing 4A.1. Itallows us to input different values of the asymmetry parameter a.

A plot of normalized guide index b versus variable V for three values of parameter a(a = 0, 8, 50) is shown in Fig. 4.3.

We finish this section with a typical simplified expression for effective index N . From adefinition of b one obtains

b(n2f − n2

s ) + n2s ≡ N2

Perform the algebraic steps

N2 =(

1 + bn2

f − n2s

n2s

)n2

s

N =√√√√1 + b

n2f − n2

s

n2s

ns

N �(

1 + b1

2

n2f − n2

s

n2s

)ns

N =(

1 + b1

2

(n f − ns)(n f + ns)

n2s

)ns

69 Fundamentals of EM theory of dielectric waveguides

0 5 10 15 200

0.2

0.4

0.6

0.8

1

V

b

ν=0

ν=1

ν=2

a=0 a=50

a=0 a=50a=0 a=50

Fig. 4.3 Normalized b − V diagram of a planar waveguide for various degrees of asymmetry.

Finally, one finds

N ≈ ns + 1

2b(n f − ns

)(4.13)

4.2 Fundamentals of EM theory of dielectric waveguides

4.2.1 General discussion

Assuming time-harmonic fields and using constitutive relations, the source-free Maxwellequations are

∇ × E = − jωμH (4.14)

∇ × H = jωεE (4.15)

Boundary conditions were derived in Chapter 3. Separate fields into transversal and longi-tudinal components as

E = Et + Ez, H = Ht + Hz (4.16)

where

Et = [Ex, Ey, 0

]is the transverse part and

Ez = [0, 0, Ez]

70 Slab waveguides

is the longitudinal part. Also

∇ = ∇t + az∂

∂z, az = [0, 0, 1] (4.17)

where az is a unit vector along z-direction. Substituting (4.16) and (4.17) into Eqs. (4.14)and (4.15), one obtains

∇t × Et = − jωμHz (4.18)

∇t × Ht = jωεEz (4.19)

∇t × Ez + az × ∂Et

∂z= − jωμHt (4.20)

∇t × Hz + az × ∂Ht

∂z= jωεEt (4.21)

Modes in a waveguide are characterized by a dielectric constant

ε(x, y) = ε0n2 (x, y) (4.22)

where n (x, y) is the refractive index profile in the transversal plane. Write the fields as

E (x, y, z) = Eν (x, y) e− jβν z (4.23)

H (x, y, z) = Hν (x, y) e− jβν z

where we have introduced index ν which label modes and βν is the propagation constantof the mode ν. After substitution of Eqs. (4.23) into (4.18)–(4.21) one obtains

∇t × Etν (x, y) = − jωμHzν (x, y) (4.24)

∇t × Htν (x, y) = jωεEzν (x, y) (4.25)

∇t × Ezν (x, y) − jβνaz × Etν (x, y) = − jωμHtν (x, y) (4.26)

∇t × Hzν (x, y) − jβνaz × Htν (x, y) = jωεEtν (x, y) (4.27)

From the analysis of the above equations, the existence of several types of modes can berecognized. Those will be described in more details later on. Generally, the main modes are:

• guided modes (bound states) – discrete spectrum of βν

• radiation modes – belong to continuum• evanescent modes-βν = − jαν ; they decay as exp (−ανz).

4.2.2 Explicit form of general equations

Using the following general formulas

∇t × Et =

∣∣∣∣∣∣∣ax ay az∂∂x

∂∂y 0

Ex Ey 0

∣∣∣∣∣∣∣∇t × Ez =

∣∣∣∣∣∣∣ax ay az∂∂x

∂∂y 0

0 0 Ez

∣∣∣∣∣∣∣

71 Wave equation for a planar wide waveguide

(TE mode)(TM mode)Hy

Ey

x

y

z

Fig. 4.4 Planar wide waveguide.

az × Et =

∣∣∣∣∣∣∣ax ay az

0 0 1Ex Ey 0

∣∣∣∣∣∣∣where ax, ay, az are unit vectors along the corresponding directions, we write generalEqs. (4.24)–(4.27) in expanded form (dropping mode index)[

0, 0,∂Ey

∂x− ∂Ex

∂y

]= − jωμ [0, 0, Hz] (4.28)[

0, 0,∂Hy

∂x− ∂Hx

∂y

]= jωε [0, 0, Ez] (4.29)[

∂Ez

∂y,−∂Ez

∂x, 0

]− jβ

[−Ey, Ex, 0] = − jωμ

[Hx, Hy, 0

](4.30)[

∂Hz

∂y,−∂Hz

∂x, 0

]− jβ

[−Hy, Hx, 0] = jωε

[Ex, Ey, 0

](4.31)

In the following sections the previous general equations will be used to analyse specificsituations.

4.3 Wave equation for a planar wide waveguide

For the wide waveguide where its dimension in the y-direction is much larger than thickness,the field configuration along that direction remains approximately constant, see Fig. 4.4.Therefore, we can consider only confinement in the x-direction and set

∂

∂y= 0

Also, the refractive index assumes only x-dependence

n = n (x)

Such a waveguide supports two modes:

• transverse electric TE, where Ey �= 0 and Ex = Ez = 0• transverse magnetic TM, where Hy �= 0 and Hx = Hz = 0.

72 Slab waveguides

From general formulas given by Eqs. (4.28)–(4.31), one obtains a set of three equationsdescribing TE modes and also three equations describing TM modes. Those are writtenbelow for both types of modes separately.

TE modesTo describe TE modes we use only equations which involve Ey and its derivatives. One

observes that when Ex = Ez = 0, from Eq. (4.30) Hy = 0. Equations describing thosemodes are

βEy = −ωμHx (4.32)

∂Hz

∂x+ jβHx = − jωεEy (4.33)

∂Ey

∂x= − jωμHz (4.34)

The last step will be to eliminate Hx and Hz. We will differentiate the third equation, thensubstitute for ∂Hz

∂x using the second equation and finally eliminate Hx using first equation.We obtain a wave equation for TE modes

∂2Ey

∂x2= (

β2 − n2k2)

Ey (4.35)

where k = ωc = ω

√ε0μ0.

TM modesWe proceed in a similar way as for the TE modes. From Eq. (4.31) when Hx = Hz = 0,

one obtains that Ey = 0. We keep only equations which involve Hy and its derivatives.Equations which describe TM modes are

βHy = ωεEx

∂Hy

∂x= jωεEz

∂Ez

∂x+ jβEx = jωμHy

We will use the fact that ε = ε0n2 and eliminate Ex and Ez. Here one must be morecareful because n2 can be x dependent. From the first equation we determine Ex and fromthe second equation we determine Ez. Substituting the results into the third equation givesthe wave equation for TM mode

n2 ∂

∂x

(1

n2∂x

∂Hy

∂x

)= (

β2 − n2k2)

Hy (4.36)

4.4 Three-layer symmetrical guiding structure (TE modes)

We will analyse the three-layer symmetrical structure for TE modes. The structure is asshown in Fig. 4.4. The details are illustrated in Fig. 4.5. A film layer of thickness 2a and

73 Three-layer symmetrical guiding structure (TE modes)

x

z

y

nf

n0

n0

0

a

−a

Fig. 4.5 Planar three-layer symmetric waveguide.

refractive index n f is surrounded by a substrate and cladding layers have refractive indicesequal to n0.

We introduce the following notation:

κ2f = n2

f k2 − β2 (4.37)

and

γ 2 = β2 − n20k2 (4.38)

Here, γ is determined by the value of refractive index around the film. In general, γ assumesvalues which are dictated by the values of refractive indices in the cladding and substrate.Therefore, one deals with more than one value of γ .

The wave equation is

d2Ey(x)

dx2= (

β2 − n2k2)

Ey(x)

We discuss two separate solutions: odd and even modes.

Odd modes:We assume the following guiding solution:

Ey(x) = Ace−γ (x−a), a<x (cladding)

Ey(x) = B sin κ f x, −a < x < a (film)

Ey(x) = Aseγ (x+a), x<-a (substrate)

(4.39)

At the interfaces, i.e. for x = ±a, the field Ey and its derivative dEy

dx must be continuous.The continuity at x = a gives the following equations:

Ac = B sin κ f a

−γ Ac = κ f B cos κ f a

From above equations, one obtains

−γ = κ f cot κ f a (4.40)

The continuity at x = −a gives an identical equation. By introducing variable

y = κ f a

Eq. (4.40) can be written as

−y cot y = γ a

74 Slab waveguides

y23

21 Rπ π

g1 g

2 g3

h

Fig. 4.6 Graphical solution of the transcendental equation for even modes.

From the definition for γ we obtain

−γ a = a√

n2k2 − β2 =√

a2n2k2 − a2β2 (4.41)

Similarly, from the definition for κ f we obtain

a2κ2f = a2nn2

f k2 − a2β2

Determining a2β2 from the above and substituting into Eq. (4.41), one finds

−y cot y =√

R2 − y2 (4.42)

where we have defined

R2 = a2k2(

n2f − n2

)(4.43)

Transcendental Eq. (4.42) must be solved numerically.

Even modes:For even modes we assume the following guiding solution:

Ey(x) = Ace−γ (x−a), a < x (cladding)

Ey(x) = B cos κ f x, −a < x < a (film)

Ey(x) = Aseγ (x+a), x < −a (substrate)

(4.44)

The remaining steps are done in the exactly same way as for odd modes. The resultingtranscendental equation for propagation constant is

y tan y =√

R2 − y2 (4.45)

The function R is defined by Eq. (4.43). Eq. (4.45) is represented in Fig. 4.6. Intersectionsbetween line h which represents the right hand side of Eq. (4.45) and lines g′s whichrepresent the left hand side of Eq. (4.45), determine propagation constants. Those must befound numerically. For numerical searches, the following functions are introduced:

feven(y) = y tan y −√

R2 − y2 (4.46)

and

fodd(y) = −y cot y −√

R2 − y2 (4.47)

75 Modes of the arbitrary three-layer asymmetric planar waveguide in 1D

The above functions are used in numerical search, since e.g. feven(y1) = 0 correspondsto the solution y1 from which propagation constant β1 is determined. The algorithm isdescribed in the next section.

4.4.1 The algorithm

The following main steps constitute the numerical algorithm.1. Search intervals for even and odd functions are formed as follows:even functions

yi = nπ, y f = min(

nπ + π

2− 10−3, R

)odd functions

yi = π

2+ nπ, y f = min

((n + 1)π − 10−3, R

)where yi and y f are initial and final values of the appropriate search interval. The smallconstant (10−3) has been introduced to avoid working close to singularity.

2. Search is done for zeros in each interval.3. Even and odd solutions are obtained independently in separate searches.4. The solutions are found using MATLAB function fzero() as

ytemp = f zero(

f unc,[yi, y f

])where the first argument contains search function and the second one contains interval forsearch.

4.5 Modes of the arbitrary three-layer asymmetric planarwaveguide in 1D

Consider the modified version of the structure shown in Fig. 4.5. The structure has differentvalues of refractive indices of substrate and cladding and it is known as asymmetric planarwaveguide. Here, nc signifies refractive index of cladding, n f of film, and ns that of substrate.For an asymmetric slab, nc �= ns. Define the following quantities:

κ2c = n2

ck2 − β2 ≡ −γ 2c

κ2f = n2

f k2 − β2 (4.48)

κ2s = n2

s k2 − β2 ≡ −γ 2s

where γi describes transverse decay and κi contains propagation constants. i takes thevalues c, f , s as appropriate. In the next subsection we discuss TE modes for this three-layer structure.

76 Slab waveguides

4.5.1 TE modes

For guided TE modes the following solutions exist:

Ey(x) = Ace−γc(x−a) a < x (cover)Ey(x) = A cos κ f x + B sin κ f x −a < x < a (film)Ey(x) = Aseγs(x+a) x < −a (substrate)

(4.49)

Derivatives are determined as follows:

dEy(x)

dx= −γcAce−γc(x−a) a < x (cover)

dEy(x)

dx= −κ f A cos κ f x + κ f B sin κ f x −a < x < a (film)

dEy(x)

dx= γsAse

γs(x+a) x < −a (substrate)

(4.50)

Boundary conditions dictate that

Ey anddEy(x)

dxare continuous for x = a and for x = −a (4.51)

When applying boundary conditions for Ey and dEy(x)

dx , we get the following equations:

for x = −a A cos κ f a − B sin κ f a = As

κ f A sin κ f a + κ f B cos κ f a = γsAs

for x = a Ac = A cos κ f a + B sin κ f a−γcAc = −κ f A sin κ f a + κ f B cos κ f a

The above equations can be written in a matrix form⎡⎢⎢⎣cos κ f a −sin κ f a −1 0

κ f sin κ f a κ f cos κ f a −γs 0cos κ f a sin κ f a 0 −1

−κ f sin κ f a κ f cos κ f a 0 γc

⎤⎥⎥⎦⎡⎢⎢⎣

ABAs

Ac

⎤⎥⎥⎦ = 0 (4.52)

For the above homogeneous system to have nontrivial solution, the main determinant shouldvanish. ∣∣∣∣∣∣∣∣

cos κ f a −sin κ f a −1 0κ f sin κ f a κ f cos κ f a −γs 0cos κ f a sin κ f a 0 −1

−κ f sin κ f a κ f cos κ f a 0 γc

∣∣∣∣∣∣∣∣ = 0 (4.53)

The above determinant is evaluated as follows. Let us expand it over last column and weget

γc

∣∣∣∣∣∣cos κ f a −sin κ f a −1

κ f sin κ f a κ f cos κ f a −γs

cos κ f a sin κ f a 0

∣∣∣∣∣∣+∣∣∣∣∣∣

cos κ f a −sin κ f a −1κ f sin κ f a κ f cos κ f a −γc

−κ f sin κ f a κ f cos κ f a 0

∣∣∣∣∣∣ = 0

Evaluating both determinants, we obtain

sin2 κ f a − cos2 κ f a + κ f

γcsin κ f a cos κ f a − γs

κ fsin κ f a cos κ f a = 0

77 Modes of the arbitrary three-layer asymmetric planar waveguide in 1D

which can be expressed as

tan2 κ f a − 1 + κ f

γctan κ f a − γs

κ ftan κ f a = 0 (4.54)

It is the general equation for a three-layer asymmetric waveguide. For a symmetricwaveguide

γs = γc = γ

and one can write Eq. (4.54) as(tan κ f a − γ

κ f

)(tan κ f a + κ f

γ

)= 0 (4.55)

which describes even and odd modes discussed in the previous section (symmetric modes).Note: The above modes are not normalized for power. Power P carried by a mode per

unit guide width is determined as follows:

P = −2

+∞∫−∞

dxEyHx

= 2β

ωμ

+∞∫−∞

dxE2y

= N

√ε0

μ0E2

f · he f f

= E f · Hf · he f f (4.56)

where he f f ≡ 2a + 1γs

+ 1γc

is the effective thickness of the waveguide.

4.5.2 Field profiles for TE modes

Analysis of the general determinant given by Eq. (4.54) for the asymmetric waveguideis complicated. If one wants to obtain formulas suitable for numerical evaluations andalso suitable for obtaining field profiles, a better approach is to eliminate all constantsappearing in Eq. (4.49). Thus, one will have a field profile expressed in terms of oneconstant only, say As. One then needs to determine derivatives which should be continuousacross interfaces. From continuity of derivatives, the relevant transcendental equation usedto obtain propagation constants is obtained.

We consider asymmetric structure where the substrate-film discontinuity is at x = 0 andfilm-cover interface is located at x = h. From continuity of the T E field at x = 0 and x = hone finds

Ac = A cos κ f h + B sin κ f h (4.57)

and

As = A (4.58)

78 Slab waveguides

Table 4.1 Asymmetric three-layer waveguide.

Refractive index Thickness

nc = 1.40 —n f = 1.50 5 µmns = 1.45 —

From continuity of derivatives at the above points

−γcAc = −κ f A sin κ f h + κ f B cos κ f h (4.59)

and

γsAs = κ f B (4.60)

From Eqs. (4.58) and (4.60) the constants A and B can be expressed in terms of As.Substitution of those expressions into Eq. (4.57) gives constant Ac in terms of As. Usingthose results, we can replace constants A, B and Ac in Eq. (4.49). The resulting equationsare

Ey(x) = As

(cos κ f h + γs

κ fsin κ f h

)exp [−γc (x − h)] h < x

Ey(x) = As

(cos κ f x + γs

κ fsin κ f x

)0 < x < h

Ey(x) = As exp (γsx) x < 0

(4.61)

From the above, the derivatives with respect to x are

E′y(x) = −Asγc

(cos κ f h + γs

κ fsin κ f h

)exp [−γc (x − h)] h < x

E′y(x) = As

(−κ f sin κ f x + γs cos κ f x)

0 < x < h

E ′y(x) = Asγs exp (γsx) x < 0

(4.62)

Applying continuity of derivatives at x = h, one finds

−γc

(cos κ f h + γs

κ fsin κ f h

)= −κ f sin κ f h + γs cos κ f h

From the above, it follows the following transcendental equation:

tan κ f h = γs + γc

κ f − γcγs/κ f(4.63)

The above equation is used in numerical search for propagation constants. A typical plotof Eq. (4.63) is shown in Fig. 4.7. Using those propagation constants, from Eq. (4.61) onefinds field profiles.

Example We consider the three-layer asymmetric planar structure described in Table 4.1(from [1]) operating with light of wavelength λ = 1 µm.

79 Multilayer slab waveguides: 1D approach

Table 4.2 Propagation constants for a three-layer asymmetric structure defined in Table 4.1.

Mode Propagation constant β(µm−1)

TE0 9.40873TE1 9.36079TE2 9.28184TE3 9.17521

8.9 9 9.1 9.2 9.3 9.4 9.5−10

−8

−6

−4

−2

0

2

4

6

8

10

β

Sear

ch fu

nctio

n

Fig. 4.7 Graphical plot of Eq. (4.63) for the three-layer asymmetric waveguide defined in Table 4.1.

The structure is analysed with MATLAB code from Appendix 4A.2.1. The resultingpropagation constants are summarized in Table 4.2. There is a good agreement with pub-lished results [1]. Field profiles for two modes are plotted in Fig. 4.8.

4.6 Multilayer slab waveguides: 1D approach

In this section we discuss slab waveguides consisting of more than three layers.Multilayers require the repeated applications of boundary conditions at the layer

80 Slab waveguides

0 10 20 300

1

2

3

4

5

x (microns)

TE e

lect

ric fi

eld

0 10 20 30−4

−2

0

2

4

x (microns)

TE e

lect

ric fi

eld

Fig. 4.8 Field profiles for an asymmetric three-layer structure. Fundamental modeTE0 (left). ModeTE1 (right).

ns

n1

n2

n3

n4

nc

x1

d1

x2d2

x3d3

x4d4

x5d5

x6d6

x7

thickness

refractiveindex

x

z

x

E y

2

Fig. 4.9 Multilayer guiding structure. Electric field profile is also shown.

interfaces. Schematically, the structure is as shown in Fig. 4.9. We make the followingassumptions:

• 1D problem• n = n(x) is the refractive index• ∂

∂y = 0

Discussion of TE and TM modes is done separately. We start with TE modes.


4.6.1 TE mode

The initial equations were derived previously, Eqs. (4.32)–(4.34). We write them here foreasy reference:

∂Ey

∂x= − jωμ0Hz

−βEy = ωμ0Hx (4.64)

∂Hz

∂x+ jβHx = − jωμ0ε0εrEy

The above system can be combined together to obtain a wave equation, as

∂2Ey

∂x2= − jωμ0

∂Hz∂x from first Eq.

= − jωμ0(− jβHx − jωε0εrEy

)from third Eq.

= − jωμ0

{− jβ

(−βEy

ωμ0

)− jωε0εrEy

}from second Eq.

= β2Ey−ω2μ0ε0εrEy

Finally, one writes wave equation for ‘i’ layer as

∂2

∂x2Eyi(x) = (

β2 − k20n2

i

)Eyi(x) (4.65)

where k20 = μ0ε0ω

2 and εri = n2i . Introduce

κ2i = k2

0n2i − β2 (4.66)

The solution to the wave equation is

Eyi(x) = Aie− jκi(x−xi−1) + Bie

jκi(x−xi−1) (4.67)

At the interfaces of layers, the boundary condition of TE mode are

Ei(x)∣∣x=xi−1 = Ei−1(x)

∣∣x=xi−1 (4.68)

∂Ei(x)

∂x

∣∣x=xi−1 = ∂Ei−1

∂x

∣∣x=xi−1 (4.69)

It is more convenient to work with another set of variables Ui(x) and Vi(x) defined as

Ui(x) = Eyi(x) (4.70)

Vi(x) = ωμ0Hzi(x) (4.71)

From Eqs. (4.64) one obtains equations for new variables. Directly from the first ofEqs. (4.64)

dUi(x)

dx= − jVi(x) (4.72)

Using the second equation to eliminate Hx in the third of Eqs. (4.64), one has

∂Hz

∂x+ jβ

(−β)

ωμ0Ey = − jωμ0ε0εrEy

82 Slab waveguides

or

ωμ0∂Hz

∂x− jβ2Ey = − jω2μ0ε0εrEy

= − jk20n2

i Ey

For layer i − th

dVi(x)

dx− jβ2Ui(x) = − jk2

0n2i Ui(x)

ordVi(x)

dx= − jκ2

i Ui(x) (4.73)

Eqs. (4.72) and (4.73) can be written in a matrix form as

d

dx

[Ui(x)

Vi(x)

]=[

0 − j− jκ2

i 0

] [Ui(x)

Vi(x)

](4.74)

From Maxwell’s equations one can verify the following relation:

jdUi(x)

dx= j

∂Eyi(x)

∂x= ωμ0Hzi(x) = Vi(x)

Using the above relation, the solutions of Eqs. (4.74) are

Ui(x) = Aie− jκi(x−xi ) + Bie

jκi(x−xi )

Vi(x) = jdUi(x)

dx= κi

{Aie

− jκi(x−xi ) − Biejκi(x−xi )

}They can be written in a matrix form as[

Ui(x)

Vi(x)

]=[

e− jκi(x−xi ) e jκi(x−xi )

κie− jκi(x−xi) −κie jκi(x−xi )

] [Ai

Bi

](4.75)

From the above, for xi one finds[Ui(xi)

Vi(xi)

]=[

1 1κi −κi

] [Ai

Bi

]Inverting the last equation gives the expression for coefficients in terms of field and itsderivative [

Ai

Bi

]= 1

2

[1 1

jκi

1 − 1jκi

][Ui(xi)

Vi(xi)

]Substituting the last equation into (4.75) gives[

Ui(x)

Vi(x)

]=[

e− jκi(x−xi ) e jκi(x−xi )

κie− jκi(x−xi ) −κie jκi(x−xi)

]1

2

[1 1

jκi

1 − 1jκi

][Ui(xi)

Vi(xi)

](4.76)

= ←→Ti (xi)

[Ui(xi)

Vi(xi)

]


Explicitly, propagation matrix←→Ti (xi) at point xi is

←→Ti (xi) =

[cos κi(x − xi) − j

κisin κi(x − xi)

− jκi sin κi(x − xi) cos κi(x − xi)

](4.77)

Equation (4.76) ‘propagates’ values of the fields from point xi to an arbitrary point x withinlayer i.

4.6.2 Propagation constant

A propagation constant is obtained if we propagate fields across all interfaces. First considerpropagation between two consecutive interfaces, that is propagation from xi location to xi+1.One finds from Eq. (4.76)[

Ui+1(xi+1)

Vi+1(xi+1)

]=[

cos κi(xi+1 − xi) − jκi

sin κi(xi+1 − xi)

− jκi sin κi(xi+1 − xi) cos κi(xi+1 − xi)

][Ui(xi)

Vi(xi)

]Let us define the thickness of layer i as

di = xi+1 − xi

The propagation matrix which propagates fields from xi to xi+1 is called←→Mi and has the

form

←→Mi =

[cos κidi − j

κisin κidi

− jκi sin κidi cos κidi

](4.78)

Using the above formalism, we can now propagate fields through the entire structure fromsubstrate to a cladding region as follows:[

Uc

Vc

]=

N∏i=1

←→Mi

[Us

Vs

]

= ←→M

[Us

Vs

]≡[

m11 m12

m21 m22

] [Us

Vs

](4.79)

In the above we have introduced the 2 × 2 matrix←→M with elements m11, m12, m21, m22

which plays central role in the numerical process of determining propagation constants.From the above equation one obtains

Uc = m11Us + m12Vs (4.80)

Vc = m21Us + m22Vs (4.81)

In the substrate and cladding, the fields decay exponentially (Fig. 4.10). Therefore, theexpressions for fields in the cladding and substrate regions are

84 Slab waveguides

substrate waveguide cladding

x i+1xix1 xN

Fig. 4.10 Schematic of the field distribution. Exponential decay in the substrate and cladding regions is shown.

• for x < x1(substrate)

Us(x) = As exp[γs(x − x1)] (4.82)

Vs(x) = jγsAs exp[γs(x − x1)] (4.83)

• for x > xN (cladding)

Uc(x) = Ac exp[−γc(x − xN )] (4.84)

Vc(x) = − jγcAs exp[−γc(x − xN )] (4.85)

In the above expressions we defined

γ 2i = β2 − k2

0n2i (4.86)

γi = jκi

From expressions (4.82) and (4.83) one obtains at point x1

Us(x1) = As (4.87)

Vs(x1) = jγsAs (4.88)

In a similar way for point xN one has

Uc(xN ) = Ac (4.89)

Vc(xN ) = − jγcAc (4.90)

Substituting the above results into Eqs. (4.80) and (4.81), one obtains the expressions whichare used to find propagation constants employing numerical procedure

γcγsm12 − m21 = j(γcm11 + γsm22) (4.91)

Values of matrix elements m11, m12, m21, m22 are obtained numerically for a particularguiding structure.

4.6.3 Electric field

Once propagation constants are found, one can obtain profile of electric field for each mode.To do this one uses Eq. (4.76), which after multiplication of matrices is[

Ui(x)

Vi(x)

]=[

cos κi(x − xi) − jκi

sin κi(x − xi)

− jκi sin κi(x − xi) cos κi(x − xi)

][Ui(xi)

Vi(xi)

](4.92)

85 Examples: 1D approach

where for layer i xi ≤ x ≤ xi+1. The above determines electric field at an arbitrary locationwithin layer i once the values of fields are known at point xi. In practice one assumes somevalue of electric field at point x1 and using expression (4.92) determines field within alllayers. The above procedure will be applied to analyse several guiding structures in the nextsection. Before this, we will formulate basic equations for TM modes.

4.6.4 TMmodes

The analysis for TM modes follows exactly the same path as for TE modes. Here we outlinebasic equations only. For TM modes one has the following relations:

Ey = Hz = Hx = 0 (4.93)

For each layer one has

βHy = ωεEx

∂Hy

∂x= jωεEz

∂Ez

∂x+ jβEx = jωμHy (4.94)

Define variables

U = Hy

V = ωε0Ez (4.95)

They fulfill the following equations:

dU

dx= jn2V

dV

dx= j

(k2 − β2

n2

)U (4.96)

From this point one follows a similar approach as for TE modes. The remaining analysis isleft as a project.

4.7 Examples: 1D approach

In this section we will consider several waveguiding structures described in literature andapply our formalism to analyse those structures. The tests were performed in searching forpropagation constant β and then determining the profile of electric field. Our tests wereperformed for several structures analysed in References [2], [3], [4] and [5].

We attempted to find all modes. Excellent agreement was found as compared with thepublished results.

86 Slab waveguides

Table 4.3 Geometry of a four-layer lossless dielectric waveguide [3].

Layer Refractive index Thickness Description

D6 n6 = 1.00 1 µm CladdingD5 n5 = 1.66 0.50 µmD4 n4 = 1.53 0.50 µmD3 n3 = 1.60 0.50 µmD2 n2 = 1.66 0.50 µmD1 n1 = 1.50 1 µm Substrate

Table 4.4 Propagation constants for a four-layer lossless dielectric waveguide defined in Table 4.3.

Mode Propagation constant

T E0 1.622728682325434 + 0.000000000000441iT E1 1.605275698094530 + 0.000000000000014iT E2 1.557136152293222 + 0.000000000000436iT E3 1.503587112022723

The list of MATLAB files is provided in Table 4.121. All MATLAB files are listed inAppendix 4A.3.

4.7.1 Four-layer lossless waveguide

The first tested structure is the one considered by Chilwell and Hodgkinson [3] and also byChen et al. [4]. It is a four-layer lossless dielectric structure shown in Table 4.3. Wavelengthis λ = 0.6328 µm. Determined propagation constants for all four modes are summarizedin Table 4.4. They show an excellent agreement with the results published in Ref. [3] and[4].

Electric field for fundamental TE-mode has been calculated using mesh points generatedby function mesh x.m. The resulting profile of TE fundamental mode is shown in Fig. 4.11.

4.7.2 Six-layer lossy waveguide

The second structure tested by our program is defined in Table 4.5, from [4]. It is a six-layer lossy waveguide structure operating at the wavelength λ = 1.523 µm. Propagationconstants for this structure are summarized in Table 4.6. Obtained results show an excellentagreement with the published data.

The TE electric field distribution of the fundamental mode for this structure is shown inFig. 4.12.

1 I acknowledge the help of my former student, Mr L. Glowacki, in developing some of those routines.

87 Examples: 1D approach

Table 4.5 Geometry of a six-layer lossy dielectric waveguide.

Layer Refractive index Thickness (µm) Description

D8 n8 = 1.00 1 CladdingD7 n7 = 3.38327 0.10D6 n6 = 3.39614 0.20D5 n5 = 3.5321– j0.08817 0.60D4 n4 = 3.39583 0.518D3 n3 = 3.22534 1.60D2 n2 = 3.16455 0.60D1 n1 = 3.172951 1 Substrate

Table 4.6 Results of propagation constants for six-layer lossy dielectric waveguide.

Mode Propagation constant

TE0 3.460829693510364 + 0.072663342917385iTE1 3.316707802046375 + 0.023275817588121iTE2 3.208555428734457 + 0.012782067986633iTE3 3.195490593396514 + 0.012585955654404i

0 1 2 3 40

0.2

0.4

0.6

0.8

1

x (microns)

TE

ele

ctri

c fi

eld

Fig. 4.11 Field profile of fundamental TE mode for four-layer lossless structure.

4.7.3 Structure by Visser

This structure was described by Visser et al. [5]. It is summarized in Table 4.7.The results of calculations of propagation constants for TE modes are summarized in

Table 4.8. They are identical to the values published by Visser et al. [5]. The profile of theelectric field for mode T E1 is shown in Fig. 4.13.

88 Slab waveguides

Table 4.7 Geometry of a six-layer lossy dielectric waveguide described by Visser et al. [5].

Layer Thickness (µm) Refractive index

clad 1.03 0.6 3.40 − 0.002i2 0.4 3.60 + 0.010i1 0.6 3.40 − 0.002isubstrate 1.0

Table 4.8 Propagation constants obtained for the structure shown in Table 4.7.

Layer Propagation constant

T E0 3.503443332950034 − 0.007103000978683iT E1 3.337286858209064 + 0.000229491103731iT E2 3.251685206983383 + 0.000530514779912iT E3 3.104251421414573 − 0.001337986339752iT E4 2.878636779881233 + 0.000173729890361iT E5 2.628139320470185 − 0.001548644353782iT E6 2.243951362601185 − 0.000708377958009iT E7 1.768190960412424 − 0.001353217183860iT E8 1.074262026525778 − 0.002457891473570i

0 2 4 60

0.2

0.4

0.6

0.8

1

x (microns)

TE

ele

ctri

c fi

eld

Fig. 4.12 Field profile of fundamental TE mode for a six-layer lossy waveguide.

4.8 Two-dimensional (2D) structures

In the previous sections we have discussed one-dimensional guiding structures, alsoknown as planar waveguides. However, in practical devices the waveguides are essentiallytwo-dimensional (2D) In those structures, refractive index n(x, y) depends on transversecoordinate x and lateral coordinate y. Two examples of popular structures are shown in

89 Two-dimensional (2D) structures

0 1 2 30

0.2

0.4

0.6

0.8

1

x (microns)

TE e

lect

ric fi

eld

Fig. 4.13 Field profile ofT E1 mode for structure discussed by Visser et al. [5].

n1

n2d

w

n1

n2

dw

(a) (b)

n1

Fig. 4.14 Popular types of 2D waveguides. (a) Buried-channel waveguide. (b) Rib waveguide.

Fig. 4.14. They exhibit distinctive features of their refractive index. The structures shownare known as the buried channel waveguide and the rib waveguide.

In what follows we will discuss one of the methods of obtaining approximate solutionfor propagating mode, known as the effective index method.

4.8.1 Effective index method

To illustrate the effective index method [6], let us start with a scalar wave equation in 2Dwhich is

∂2φ(x, y)

∂x2+ ∂2φ(x, y)

∂y2+ k2

0

[εr(x, y) − N2

e f f

]φ(x, y) = 0 (4.97)

Here N2e f f is the effective index which we want to determine. Assume that there is no

‘interaction’ between variables x and y. This allows us to separate fields as [7]

φ(x, y) = X (x) · Y (y)

Substitute postulated solution into wave equation, evaluate the relevant derivatives anddivide both sides of the resulting equation by X (x) · Y (y). One obtains

1

X (x)X ′′(x) + 1

Y (y)Y ′′(y) + k2

0

[εr(x, y) − N2

e f f

]= 0 (4.98)

90 Slab waveguides

n1

n 2

n 3

2N

2N

3N

3N

N1

N1

The above is equivalent to

w

x

Step 3

Step 2

Step 1

n1

a1a 2

w

n 3

n 2

x

y

a1

n1

n 2 a 2

n 3y

n1

n 2 a 2

n 3y

y

Fig. 4.15 Illustration of the effective index method.

Introduce separation constant k20λ

2 as

1

Y (y)Y ′′(y) + k2

0εr(x, y) = k20λ

2 (4.99)

The remaining terms are

1

X (x)X ′′(x) − k2

0N2e f f = −k2

0λ2 (4.100)

Based on the above theory it is possible to design a method, known as effective indexmethod to determine effective index N2

e f f . The method (see Fig. 4.15) consists of severalsteps:

1. Replace the 2D waveguide with a combination of 1D waveguides along x-axis.2. Solve each of the 1D problems separately by calculating the effective index along y axis

and obtain Ne f f in each case (for our structure three cases).3. Construct a new ‘effective’ 1D waveguide (along x-axis) which will model the original

2D waveguide. The new effective waveguide has the values of refractive indices asindicated.

4. Determine an effective index by solving the 1D waveguide constructed in Step 3 alongthe x-axis.

91 Two-dimensional (2D) structures

Table 4.9 Numerical parameters used in an example of the effective index method.

Wavelength Refractive indices Geometry

λ = 0.8 µm nc = 1 h = 1.8 µmn f = 2.234 d = 1 µmns = 2.214 w = 2 µm

hd

w

nc

ns

nf

x

y

Fig. 4.16 Structure used as an example of the application of the effective index method.

An exampleWe illustrate the effective index method with an example can be found in [8]. The

structure is shown in Fig. 4.16 and the numerical parameters in Table 4.9.The method proceeds with the following four steps:

1. Determine normalized thicknesses using the definition of V

Vf = k · h√

n2f − n2

s , Vd = k · d√

n2f − n2

s

Using the values from Table 4.9, one finds

Vf = 4.2, Vd = 2.3

2. Obtain values of b using b − V diagram. From the diagram, for T E modes we obtain(for a = ∞)

b f = 0.65, bd = 0.2

Effective indices Nf and Nd are determined using

N2f ,d = n2

s + b f ,d

(n2

f − n2s

)One obtains

Nf = 2.227, Nd = 2.218

3. The same structure along the y-direction is approximated as the one-dimensional, usingthe previous values. The V number is determined using

Veg = k · w

√N2

f − N2d = kw

√(n2

f − n2s

) (b f − bd

)(4.101)

92 Slab waveguides

One obtains

Veg = 3.14, beg = 0.64 (4.102)

4. Finally, the approximate effective index N = Neq of the equivalent guide is determinedusing

N ≡ Neg = N2d + beq

(N2

f − N2d

)(4.103)

For this example, we obtain N = 2.224.More examples of the applications of effective index method can be found in books by

Coldren and Corzine [9] and Liu [10].

4.9 Problems

1. Find the effective refractive indices and the number of TE modes in an asymmetricwaveguide with an asymmetry parameter a = 10 and thickness h = 1.50 µm. Assumeλ0 = 0.82 µm.

2. Compute the largest thickness that will guarantee single TE mode operation of theabove waveguide.

3. Analyse the three-layer, asymmetric guiding structure. Derive a transcendental equa-tion for the propagation constant.

4. Conduct analysis of a four-layer waveguiding structure [8]. Derive equations for allelements of matrix M . Obtain an analytical formula for dispersion relation.

4.10 Projects

1. Based on theoretical results developed for the three-layer symmetrical guiding structure,write a MATLAB program which will solve transcendental equations using functionsgiven by (4.46) and (4.47). Select from the literature an appropriate waveguiding struc-ture. Use your program to determine propagation constants and profiles of electric fields.Compare your results with those from the literature.

2. Analyse TM modes for the three-layer asymmetric waveguide defined in Table 4.1. Ob-tain analytical solutions in all regions and derive an appropriate transcendental equation.Write a MATLAB program to determine propagation constants.

3. Plasmonic waveguide contains a metallic layer. The simplest cases are the three-layerstructures: metal-dielectric-metal (MDM) and dielectric-metal-dielectric (DMD). Derivea transcendental equation describing plasmonic waveguide (similar to Eq. 4.45 for asymmetrical dielectric guiding structure). Consult the paper by Kekatpure et al. [11].With the equation at hand, write a MATLAB program to determine propagation constants

93 Appendix 4A

for TM modes for the MDM structure defined in Table 4.10 [11]. Analyse the obtainedresults.

Table 4.10 Plasmonic structure.

Material Thickness (µm) Relative permittivity

gold 3 −95.92 − i10.97silica 50 2.1025silver 3 −143.49 − i9.52

4. Conduct analysis of the multilayered structure for TM modes. Derive all necessaryequations. Write a MATLAB program to determine propagation constants and fieldprofiles.


In Tables 4.11 and 4.12 we provide a list of MATLAB files created for Chapter 4 and ashort description of each function. In Table 4.12 we summarize a list of files used to analyseplanar waveguides.



4A.1 b V diagram.m Plots b-V diagram for planar waveguide4A.2.1 a3L.m Analysis for three-layer planar waveguide4A.2.2 f unc asym.m Function used by a3L.m

Table 4.12 Functions used to analyse a 1D planar waveguide.


4A.3.1 slab.m driver function (determines prop. constants and field profile)4A.3.2 lossless.m data for waveguide structure without losses4A.3.3 lossy.m data for waveguide structure with losses4A.3.4 visser.m data for structure described by Visser4A.3.5 muller.m implements Muller’s method4A.3.6 f TE.m creates transcendental equation for TE field4A.3.7 mesh x.m generates 1D mesh4A.3.8 refindex.m assigns the value of ref. index to each mesh point4A.3.9 TE field.m determines TE field for all mesh points

94 Slab waveguides

Listing 4A.1 Program b V diagram.m. MATLAB program to analyse the transverseresonance condition (4.12) and to plot a normalized guide index b versus variable V .Allows to input different values of the asymmetry parameter a.

% File name: b_V_diagram.m

% function which plots b-V diagram of a planar slab waveguide

clear all


b = linspace(0,1.0,N_max);

hold on

for nu = [0, 1, 2]

for a = [0.0, 8.0,50.0]; % asymmetry coefficient

% determine V

V1 = atan(sqrt(b./(1-b)) );

V2 = atan(sqrt((b+a)./(1.0-b)));

V3 = 1./sqrt(1.0-b);

V = (nu*pi + V1 + V2).*V3;

%

plot(V,b,’LineWidth’,1.2)

grid on

axis([0.0 20.0 0.0 1.0]) % change axis limit

end

end

box on % makes frame around plot

xlabel (’V’,’FontSize’,14);

ylabel(’b’,’FontSize’,14);


text(10, 0.96, ’\nu=0’,’Fontsize’,14)

text(10, 0.8, ’\nu=1’,’Fontsize’,14)

text(10, 0.55, ’\nu=2’,’Fontsize’,14)

%

text(0.1, 0.3, ’a=0’,’Fontsize’,14)

text(2.5, 0.3, ’a=50’,’Fontsize’,14)

%

text(3.3, 0.2, ’a=0’,’Fontsize’,14)

text(5.8, 0.2, ’a=50’,’Fontsize’,14)

%

text(6, 0.1, ’a=0’,’Fontsize’,14)

text(8.8, 0.1, ’a=50’,’Fontsize’,14)

pause

close all

95 Appendix 4A

Listing 4A.2.1 Program a3L.m. Determines propagation constants and plots field profilesfor an asymmetric three-layer planar waveguide. Uses func asym.m.

% File name: a3L.m

% Analysis for TE modes

% First, we conduct test plots to find ranges of beta where possible

% solutions exists. This is done in several steps:

% 1. Plots are done treating kappa_f as an independent variable

% 2. Ranges of kappa_f are determined where there are zeros of functions

% 3. Corresponding ranges of beta are determined

% 4. Searches are performed to find propagation constants

%

clear all

% Definition of structure

n_f = 1.50; % ref. index of film layer

n_s = 1.45; % ref. index of substrate

n_c = 1.40; % ref. index of cladding

lambda = 1.0; % wavelength in microns

h = 5.0; % thickness of film layer in microns

a_c = 2*h; % thickness of cladding region

a_s = 2*h; % thickness of substrate region

%

k = 2*pi/lambda; % wave number

kappa_f = 0:0.01:3.0; % establish range of kappa_f

beta_temp = sqrt((n_f*k)^2 - kappa_f.^2);

beta_min = min(beta_temp);

beta_max = max(beta_temp);

% Before searches, we plot search function versus beta

beta = beta_min:0.001:beta_max; % establish range of beta

N = beta./k;

ff = func_asym(beta,n_c,n_s,n_f,k,h);

plot(beta,ff)

xlabel(’\beta’,’FontSize’,22);

ylabel(’Search function’,’FontSize’,22);

ylim([-10.0 10.0])

grid on

pause

close all

% From the above plot, one must choose proper search range for each mode.

% Search numbers provided below are only for the waveguide defined above.

% For different waveguide, one must choose different ranges

% for searches.

beta0 = fzero(@(beta) func_asym(beta,n_c,n_s,n_f,k,h),[9.40 9.41])




% Plot of field profiles

96 Slab waveguides

A_s = 1.0;

thickness = h + a_c + a_s;

beta_field = beta0; % Select appropriate propagation constant for plotting

gamma_s = sqrt(beta_field^2 - (n_s*k)^2);

gamma_c = sqrt(beta_field^2 - (n_c*k)^2);

kappa_f = sqrt((n_f*k)^2 - beta_field^2);

%

% In the formulas below for electric field E_y we have shifted

% x-coordinate by a_s

% We also ’reversed’ direction of plot in the substrate region

NN = 100;

delta = thickness/NN;

x = 0.0:delta:thickness; % coordinates of plot points

x_t = 0;

for i=1:NN+1

x_t(i+1)= x_t(i) + delta;

if (x_t(i)<=a_s);

E_y(i) = A_s*exp(gamma_s*(x_t(i)-a_s));

elseif (a_s<=x_t(i)) && (x_t(i)<=a_s+h);

E_y(i) = A_s*(cos(kappa_f*(x_t(i)-a_s))+...

gamma_s*sin(kappa_f*(x_t(i)-a_s))/kappa_f);

else (a_s+h<=x_t(i)) & (x_t(i)<=thickness);

E_y(i) = A_s*(cos(kappa_f*h)+gamma_s*sin(kappa_f*h)/kappa_f)...

*exp(-gamma_c*(x_t(i)-h-a_s));

end

end

%

h=plot(x,E_y);

% add text on x-axix and y-axis and size of x and y labels

xlabel(’x (microns)’,’FontSize’,22);

ylabel(’TE electric field’,’FontSize’,22);

set(h,’LineWidth’,1.5); % new thickness of plotting lines

set(gca,’FontSize’,22); % new size of tick marks on both axes

grid on

pause

close all

Listing 4A.2.2 Function func asym.m. Search function for the asymmetric three-layerwaveguide. Used by program a3L.m.

function f = func_asym(beta,n_c,n_s,n_f,k,h)

% Construction of search function for asymmetric 3-layers waveguide

%

gamma_c = sqrt(beta.^2 - (n_c*k)^2);

gamma_s = sqrt(beta.^2 - (n_s*k)^2);

kappa_f = sqrt((n_f*k)^2 - beta.^2);

97 Appendix 4A

%

denom = kappa_f - (gamma_c.*gamma_s)./kappa_f;

f = tan(kappa_f*h) - (gamma_s+gamma_c)./denom;

Listing 4A.3.1 Program slab.m. Driver function which determines propagation constantfor fundamental TE mode and plots electric field profiles. One-dimensional planar wave-guide of an arbitrary number of layers is assumed. The list of files is shown in Table 4.12.

% File name: slab.m

% Driver function which determines propagation constants and

% electric field profiles (TE mode) for multilayered slab structure

clear all;

format long

% Input structure for analysis (select appropriate input)

%lossless;

%lossy;

visser

%

epsilon = 1e-6; % numerical parameter

TE_mode = [];

n_max = max(n_layer);

z1 = n_max; % max value of refractive index

n_min = max(n_s,n_c) + 0.001; % min value of refractive index

dz = 0.005; % iteration step

mode_control = 0;

%

while(z1 > n_min)

z0 = z1 - dz; % starting point for Muller method

z2 = 0.5*(z1 + z0); % starting point for Muller method

z_new = muller(@f_TE , z0, z1, z2);

if (z_new ~= 0)

% verifying for mode existence

for u=1 : length(TE_mode)

if(abs(TE_mode(u) - z_new) < epsilon)

mode_control = 1; break; % mode found

end

end

if (mode_control == 1)

mode_control = 0;

else

TE_mode(length(TE_mode) + 1) = z_new;

end

end

z1 = z0;

end

%

98 Slab waveguides

TE_mode = sort(TE_mode, ’descend’);

%TE_mode’ % outputs all calculated modes

beta = TE_mode(2); % selects mode for plotting field profile

x = mesh_x(d_s,d_layer,d_c,NumberMesh);

n_total = [n_s,n_layer,n_c]; % ref index for all layers

n_mesh = refindex(x,NumberMesh,n_total);

TE_mode_field = TE_field(beta,n_mesh,x,k_0);

Listing 4A.3.2 Function lossless.m. Contains data for a lossless waveguide.

% File name: lossless.m

% Contains data for lossless waveguide

% Reference:

% J. Chilwell and I. Hodgkinson,

% "Thin-films field-transfer matrix theory of planar multilayer

% waveguides and reflection from prism-loaded waveguide",

% J. Opt. Soc. Amer.A, vol.1, pp. 742-753 (1984).

% Fig.3

% Global variables to be transferred to function f_TE.m

%

global n_c % ref. index cladding

global n_layer % ref. index of internal layers

global n_s % ref. index substrate

global d_c % thickness of cladding (microns)

global d_layer % thicknesses of internal layers (microns)

global d_s % thickness of substrate (microns)

global k_0 % wavenumber

global NumberMesh % number of mesh points in each layer

% (including substrate and cladding)

n_c = 1.0;

n_layer = [1.66 1.60 1.53 1.66];

n_s = 1.5;

d_c = 0.5;

d_layer = [0.5 0.5 0.5 0.5];

d_s = 1.0;

NumberMesh = [10 10 10 10 10 10];


k_0 = 2*pi/lambda;

Listing 4A.3.3 Function lossy.m. Contains data for a lossy waveguide.

% File name: lossy.m

% Contains data for lossy waveguide.

% Reference:

99 Appendix 4A

% C. Chen et al, Proc. SPIE, v.3795 (1999)











n_c = 1.0;

n_layer = [3.16455 3.22534 3.39583 3.5321-1j*0.08817 3.39614 3.38327];

n_s = 3.172951;

d_c = 1.0;

d_layer = [0.6 1.6 0.518 0.6 0.2 0.1];

d_s = 1.0;

NumberMesh = [10 10 10 10 10 10 10 10];


k_0 = 2*pi/lambda;

Listing 4A.3.4 Function visser.m. Contains data for a five-layer waveguide described byVisser et al.

% File name: visser.m

% Contains data for five-layer waveguide with gain and losses

% Reference:

% T.D. Visser et al, JQE v.31, p.1803 (1995)

% Fig.6


%










n_c = 1.0;

n_layer = [3.40-1i*0.002 3.60+1i*0.010 3.40-1i*0.002];

n_s = 1.0;

d_s = 0.4;

d_layer = [0.6 0.4 0.6];

100 Slab waveguides

d_c = 0.5;

NumberMesh = [10 10 10 10 10];


k_0 = 2*pi/lambda;

Listing 4A.3.5 Function muller.m. This function implements Muller’s method. It is goodfor finding complex roots.

function f_val = muller (f, x0, x1, x2)

% Function implements Muller’s method

iter_max = 100; % max number of steps in Muller method

f_tol = 1e-6; % numerical parameters

x_tol = 1e-6;

y0 = f(x0);

y1 = f(x1);

y2 = f(x2);

iter = 0;

while(iter <= iter_max)

iter = iter + 1;

a =( (x1 - x2)*(y0 - y2) - (x0 - x2)*(y1 - y2)) / ...

( (x0 - x2)*(x1 - x2)*(x0 - x1) );

%

b = ( ( x0 - x2 )^2 *( y1 - y2 ) - ( x1 - x2 )^2 *( y0 - y2 )) / ...

( (x0 - x2)*(x1 - x2)*(x0 - x1) );

%

c = y2;

%

if (a~=0)

D = sqrt(b*b - 4*a*c);

q1 = b + D;

q2 = b - D;

if (abs(q1) < abs(q2))

dx = - 2*c/q2;

else

dx = - 2*c/q1;

end

elseif (b~=0)

dx = -c/b;

else

warning(’Muller method failed to find a root’)

break;

end

x3 = x2 + dx;

x0 = x1;

x1 = x2;

101 Appendix 4A

x2 = x3;

y0 = y1;

y1 = y2;

y2 = feval(f, x2);

if (abs(dx) < x_tol && abs (y2) < f_tol)

break;

end

end

% Lines below ensure that only proper values are calculated

if (abs(y2) < f_tol)

f_val = x2;

return;

else

f_val = 0;

end

Listing 4A.3.6 Function f TE.m. Using the transfer matrix approach, a transcendentalequation is created which is then used to determine a propagation constant.

function result = f_TE(z)

% Creates function used to determine propagation constant

% Variable description:

% result - expression used in search for propagation constant

% z - actual value of propagation constant

%

% Global variables:

% Global variables are used to transfer values from data functions






%

zz=z*k_0;

NumLayers = length(d_layer);

%

% Creation for substrate and cladding

gamma_sub=sqrt(zz^2-(k_0*n_s)^2);

gamma_clad=sqrt(zz^2-(k_0*n_c)^2);

%

% Creation of kappa for internal layers

kappa=sqrt(k_0^2*n_layer.^2-zz.^2);

temp = kappa.*d_layer;

%

102 Slab waveguides

% Construction of transfer matrix for first layer

cc = cos(temp);

ss = sin(temp);

m(1,1) = cc(1);

m(1,2) = -1j*ss(1)/kappa(1);

m(2,1) = -1j*kappa(1)*ss(1);

m(2,2) = cc(1);

%

% Construction of transfer matrices for remaining layers

% and multiplication of matrices

for i=2:NumLayers

mt(1,1) = cc(i);

mt(1,2) = -1j*ss(i)/kappa(i);

mt(2,1) = -1j*ss(i)*kappa(i);

mt(2,2) = cc(i);

m = mt*m;

end

%

result = 1j*(gamma_clad*m(1,1)+gamma_sub*m(2,2))...

+ m(2,1) - gamma_sub*gamma_clad*m(1,2);

Listing 4A.3.7 Function mesh x.m. Generates a one-dimensional mesh along thetransversal direction (which is taken as the x-axis) of a dielectric waveguide.

function x = mesh_x(d_s,d_layer,d_c,NumberMesh)

% Generates one-dimensional mesh along x-axis


% Input

% d_layer - thicknesses of each layer

% NumberMesh - number of mesh points in each layer

% Output

% x - mesh point coordinates

%

d_total = [d_s,d_layer,d_c]; % thicknesses of all layers

NumberOfLayers = length(d_total); % determine number of layers

delta = d_total./NumberMesh; % separation of points for all layers

%

x(1) = 0.0; % coordinate of first mesh point

i_mesh = 1;

for k = 1:NumberOfLayers % loop over all layers

for i = 1:NumberMesh(k) % loop within layer

x(i_mesh+1) = x(i_mesh) + delta(k);

i_mesh = i_mesh + 1;

end

end

103 Appendix 4A

Listing 4A.3.8 Function refindex.m. Determines the refractive index at all mesh points.

function n_mesh = refindex(x,interface,index_layer)

% Assigns the values of refractive indices to mesh points

% in all layers

% Input

% x() -- mesh points coordinates

% interface(n) -- number of mesh points in layers

% index_layer() -- refrective index in layers

% Output

% index_mesh -- refractive index for each mesh point

%

% Within a given layer, refractive index is assigned the same value.

% Loop scans over all mesh points.

% For all mesh points selected for a given layer, the same

% value of refractive index is assigned.

%

N_mesh = length(x);

NumberOfLayers = length(index_layer);

%

i_mesh = 1;

for k = 1:NumberOfLayers % loop over all layers

for i = 1:interface(k) % loop within layer

n_mesh(i_mesh+1) = index_layer(k);

i_mesh = i_mesh + 1;

end

end

Listing 4A.3.9 Function TE field.m. Determines TE optical field for all layers.

function TE_mode_field = TE_field(beta,index_mesh,x,k_zero)

% Determines TE optical field for all layers

%

% x - grid created in mesh_x.m

TotalMesh = length(x); % total number of mesh points

%

zz=beta*k_zero;

%

% Creation of constants at each mesh point

kappa = 0;

for n = 1:(TotalMesh)

kappa(n)=sqrt((k_zero*index_mesh(n))^2-zz^2);

end

%

% Establish boundary conditions in first layer (substrate).

104 Slab waveguides

% Values of the fields U and V are numbered by index not by

% location along x-axis.

% For visualization purposes boundary conditions are set at first point.

U(1) = 1.0;

temp = imag(kappa(1));

if(temp<0), kappa(1) = - kappa(1);

end

% The above ensures that we get a field decaying in the substrate

V(1) = kappa(1);

%

for n=2:(TotalMesh)

cc=cos( kappa(n)*(x(n)-x(n-1)) );

ss=sin( kappa(n)*(x(n)-x(n-1)) );

m(1,1)=cc;

m(1,2)=-1i/kappa(n)*ss;

m(2,1)=-1i*kappa(n)*ss;

m(2,2)=cc;

%

U(n)=m(1,1)*U(n-1)+m(1,2)*V(n-1);

V(n)=m(2,1)*U(n-1)+m(2,2)*V(n-1);

end

%

TE_mode_field = abs(U); % Finds Abs(E)

max_value = max(TE_mode_field);

h = plot(x,TE_mode_field/max_value); % plot normalized value of TE field

% adds text on x-axix and size of x label

xlabel(’x (microns)’,’FontSize’,22);

% adds text on y-axix and size of y label

ylabel(’TE electric field’,’FontSize’,22);



pause

close all

References


[2] C. Chen, P. Berini, D. Feng, S. Tanev, and V. P. Tzolov. Efficient and accurate numericalanalysis of multilayer planar optical waveguides in lossy anisotropic media. OpticsExpress, 7:260–72, 2000.

[3] J. Chilwell and I. Hodgkinson. Thin-films field-transfer matrix theory of planar mul-tilayer waveguides and reflection from prism-loaded waveguide. J. Opt. Soc. Amer. A,1:742–53, 1984.

105 References

[4] C. Chen, P. Berini, D. Feng, S. Tanev, and V. P. Tzolov. Efficient and accurate numericalanalysis of multilayer planar optical waveguides. Proc. SPIE, 3795:676, 1999.

[5] T. D. Visser, H. Blok, and D. Lenstra. Modal analysis of a planar waveguide with gainand losses. IEEE J. Quantum Electron., 31:1803–18, 1995.

[6] R. Maerz. Integrated Optics. Design and Modeling. Artech House, Boston, 1995.[7] M. N. O. Sadiku. Elements of Electromagnetics. Second Edition. Oxford University

Press, New York, Oxford, 1995.[8] H. Kogelnik. Theory of optical waveguides. In T. Tamir, ed., Springer Series in Elec-

tronics and Photonics. Guided-Wave Optoelectronics, volume 26, pp. 7–88. Springer-Verlag, Berlin, 1988.

[9] L. A. Coldren and S. W. Corzine. Diode Lasers and Photonic Integrated Circuits.Wiley, New York, 1995.

[10] J.-M. Liu. Photonic Devices. Cambridge University Press, Cambridge, 2005.[11] R. D. Kekatpure, A. C. Hryciw, E. S. Barnard, and M. L. Brongersma. Solving

dielectric and plasmonic waveguide dispersion relations on a pocket calculator. OpticsExpress, 17:24112–29, 2009.

5 Linear optical fibre and signal degradation

In this chapter we will discuss properties of the linear fibre, where refractive index does notdepend on field intensity. We will concentrate on the following topics:

• geometrical-optics description• wave propagation• dispersion in single-mode fibres• dispersion-induced limitations.

Due to dispersion, the width of propagating pulse will increase. In multimode fibres,broadening of optical pulses is significant, about 10 ns km−1. In the geometrical-opticsdescription, such broadening is attributed to different paths followed by different rays. Ina single mode fibre, intermodal dispersion is absent; all energy is transported by a singlemode. However, pulse broadening exists. Different spectral components of the pulse travel atslightly different group velocities. This effect is known as group velocity dispersion (GVD).

5.1 Geometrical-optics description

Consider an optical fibre whose cross-section is shown in Fig. 5.1. The correspondingchange in refractive index profiles is shown in Fig. 5.2. We illustrate two profiles: stepindex and graded index.

In the following discussion we will assume validity of a geometrical optics descriptionwhich holds in the limit λ � a, where λ is the light wavelength, and a is the core radius[1], [2].

5.1.1 Numerical aperture (NA)

To start our discussion, consider the propagation of rays entering a cylindrical fibre atdifferent angles in the plane passing through the core centre, see Fig. 5.3. Ray B will travelin the cladding region and it is known as an unguided ray, whereas ray A will stay withincore region, and it will form a guided ray.

There exists an angle θa (known as the acceptance angle) such that all rays incident froma region with refractive index n0 at angles fulfilling the relation

θ < θa

will stay within a core region (such as ray A). All rays incident at the acceptance angle θa

will travel at the boundary between core and cladding.106

107 Geometrical-optics description

jacketb

a

cladding

core

Fig. 5.1 Cross-section of a typical optical fibre.

n

n1 n2

n0

a

bn0

n2n1

a

b

n

Fig. 5.2 Typical profiles of optical fibre. Linear step (left) and graded (right).

1 2

A

A

B

B

Core index n1

n

θ θφ

0

Cladding index n2

Fig. 5.3 Acceptance angle.

The numerical aperture (NA) is defined as

NA ≡ n0 sin θa

It is a dimensionless quantity. Typical values of NA are within the 0.14–0.50 range. Eval-uation of NA follows exactly the same procedure as described in Chapter 4. The finalexpression is

NA = n1 (2�)1/2 (5.1)

In practice n1 ≈ n2 and the relative refractive index difference � is

� = (n1 + n2) (n1 − n2)

2n21

� 2n1 (n1 − n2)

2n21

= n1−n2

n1(5.2)

A typical value of parameter � for a single-mode fibre is around 0.01.


5.1.2 Multipath dispersion

A very important physical effect originates from the fact that different rays travel alongpaths of different lengths. The shortest path, for incident angle θ1 = 0 is equal to L, fibrelength; the longest path, for θ1 = θmax (which corresponds to φc) is for rays travelling thedistance L

sin φc. Time delay between both paths is �T . It can be estimated as follows:

�T = �distance

υ

=L

sin φc− L

cn1

= n1

c

(L

sin φc− L

)

= n1

c

(Ln2n1

− L

)= Ln1

c

(n1

n2− 1

)= Ln1

cn2

n1 − n2

n1

= Ln21

cn2� (5.3)

where we have used υ = cn1

for speed of light in fibre. Time �T is a measure of broadeningof an impulse.

5.1.3 Information-carrying capacity of the fibre

Next, we will estimate the information-carrying capacity of the fibre. If we introduce B asa bit rate, and L fibre length, then the measure of the information-capacity of the fibre isdetermined by a product B · L.

The estimation of this product is as follows. Let TB where TB = 1B be the time allocated

for each bit slot. We use the condition

�T < TB (5.4)

which requires that pulse broadening is smaller than time slot allocated for each bit. Let ussubstitute the previous expression for �T , and we obtain

Ln21

cn2� · B < 1 or

L · B <n2

n21

c

�(5.5)

Example Assume n1 = 1.5, n2 = 1.0. From the above, one finds L · B < 0.4 Mbs · km. On

the other hand, for � = 2 × 10−3 ⇒ B · L < 100 Mbs · km. This means that one can carry

data of 10 Mb s−1 over 10 km.

5.1.4 Loss mechanisms in silica fibre

Attenuation (losses) of optical signal is a major factor in the design of optical communicationsystems. Losses can occur at the input coupler, splices and connectors but also within the

109 Geometrical-optics description

1.6 1.81.41.21.0

0.1

0.01

0.5

0.05

510

50100

1

0.8

waveguideimperfections

infraredabsorption

ultravioletabsorption

Rayleighscattering

wavelength (microns)

loss

(dB

/km

)

Fig. 5.4 Spectrum of losses in glass. Adapted from [3] with permission from the Institution of Engineering and Technology.

fibre itself. In this section we will concentrate on losses in the fibre. A typical spectrum oflosses in modern fibre is shown in Fig. 5.4. Losses are usually expressed in decibels perunit of distance (as was mentioned in Chapter 2) according to formula

αloss = 10

Llog10

Pout

Pin(5.6)

where Pin and Pout are, respectively, the input and output optical powers for a fibre oflength L.

Loss mechanisms are classified as [4]:

• intrinsic which arise from the fundamental material properties of the glass. These can bereduced, e.g. by appropriate choice of wavelength

• extrinsic which result due to imperfections in the fabrication process. Sources of extrinsicloss include impurities or structural imperfections.

5.1.5 Intrinsic loss

In pure silica, the three sources of loss important at visible and near-infrared wavelengthsare:

• ultraviolet absorption• infrared absorption• Rayleigh scattering.

All of the three mechanisms are wavelength dependent. The first two involve two reso-nances centred in the ultraviolet and mid-infrared, see Fig. 5.4. They originate due to strongelectronic and molecular transition bands. The ultraviolet resonance is of electronic originand is centred near λ = 0.1 µm [4]. The infrared absorption originates from lattice vibra-tional modes of silica and dopants. The vibrational modes produce absorptive resonances


x

y

z

φ

zr

Fig. 5.5 Cylindrical coordinate system.

centered between 7 µm and 11 µm. The resonances for silica and germania occur at 9 µmand 11 µm, respectively [4]. Intraband absorption results in a power loss approximatelygiven by

αir = A exp(−air

λ

)[dB km−1] (5.7)

where λ is in micrometres. For GeO2 − SiO2 glass, the values of A and air are A =7.81 × 1011 dB km−1 and air = 48.48 µm [4].

Rayleigh scattering in pure fused silica can be approximated by the expression

αR = 1.7

(0.85

λ

)4

(5.8)

where λ is in micrometres and loss αR is in dB km−1.Rayleigh scattering appears when a wave travels through a medium having scattering

objects smaller than a wavelength [5]. In a glass those small objects are created as lo-cal variations in refractive index in otherwise homogeneous material. Local variations ofrefractive index in turn can be due to localized variations in a material’s density.

5.1.6 Extrinsic loss

These losses are not associated with the fundamental properties of material. There aremany sources of extrinsic losses, for example presence of additional impurities and surfaceirregularities at the core-cladding boundary. Some other related losses are due to bendingand source-fibre coupling.

5.2 Fibre modes in cylindrical coordinates

Fibre exhibits cylindrical symmetry, see Fig. 5.5. In the cylindrical coordinate system weuse the following variables: r, φ, z. Refractive index is expressed as (see Fig. 5.1)

n ={

n1, r ≤ an2, r > a

}(5.9)

where a is the radius of the central region known as core.

111 Fibre modes in cylindrical coordinates

5.2.1 Maxwell’s equations in cylindrical coordinates

In cylindrical coordinates, one has the following general relation for arbitrary vector A [6]:

∇×A = 1

r

∣∣∣∣∣∣∣ar raφ az∂∂r

∂∂φ

∂∂z

Ar rAφ Az

∣∣∣∣∣∣∣= ar

(∂Az

r∂φ− ∂Aφ

∂z

)+ aφ

(∂Ar

∂z− ∂Az

∂r

)+ az

1

r

[∂

∂r

(rAφ

)− ∂Ar

∂φ

]where ar, aφ, az are unit vectors in the corresponding directions. Vector A is expressed as

A = arAr + aφAφ + azAz

Assuming e jωt dependence for fields, the first Maxwell equation in cylindrical coordinatesis

1

r

∂Ez

∂φ− ∂Eφ

∂z= − jωμHr

∂Er

∂z− ∂Ez

∂r= − jωμHφ

1

r

[∂

∂r

(rEφ

)− ∂Er

∂φ

]= − jωμHz

Assuming additionally e jβz dependence, one finds

1

r

∂Ez

∂φ+ jβEφ = − jωμHr (5.10)

jβEr + ∂Ez

∂r= jωμHφ (5.11)

1

r

[∂

∂r

(rEφ

)− ∂Er

∂φ

]= − jωμHz (5.12)

Similarly, from the second Maxwell equation we obtain

1

r

∂Hz

∂φ+ jβHφ = jωεEr (5.13)

jβHr + ∂Hz

∂r= − jωεEφ (5.14)

1

r

[∂

∂r

(rHφ

)− ∂Hr

∂φ

]= jωεEz (5.15)

Expressing Hr from (5.10) and (5.14) and comparing both expressions, one obtains for Eφ

Eφ = − j

q2

(β

r

∂Ez

∂φ− ωμ

∂Hz

∂r

)(5.16)

where

q2 = ω2εμ − β2 (5.17)

= n2k20 − β2


Similarly, using the same procedure, from (5.11) and (5.13) one obtains for Er

Er = − j

q2

(β

∂Ez

∂r+ ωμ

r

∂Hz

∂φ

)(5.18)

In an analogous way one finds expressions for Hφ and Hr:

Hφ = − j

q2

(β

r

∂Hz

∂φ+ εω

∂Ez

∂r

)(5.19)

Hr = − j

q2

(β

∂Hz

∂r− ε

ω

r

∂Ez

∂φ

)(5.20)

Equations (5.16), (5.18), (5.19) and (5.20) express relevant components of E and H fields interms of Ez and Hz components. In the next step, we derive equations for those components.

5.2.2 Wave equations in cylindrical coordinates

From the analysis above we can observe that the z-component of the electric field Ez (andsame applies also to magnetic z-component Hz) can be considered an independent variableand that the remaining components can be expressed in terms of Ez. (The same is true forHz.) Hence, we only need to work with Ez (or Hz). For that purpose we will derive theappropriate wave equation from which we can obtain a solution for Ez. Once we have asolution for Ez we will use it to find Er and Eφ . In this way, all components of the electricfield within a circular waveguide are derived.

To derive the equation for Ez we start with Eq. (5.15). Replacing derivatives on the lefthand side by expressions (5.19) and (5.20) and rearranging terms, one finds

∂2Ez

∂r2+ 1

r

∂Ez

∂r+ 1

r2

∂2Ez

∂φ2+ q2Ez = 0 (5.21)

where q2 is given by Eq. (5.17).Similarly, we can derive an equation for Hz. For the last step we will use Eq. (5.12). We

replace Eφ and Er and obtain a wave equation for Hz in cylindrical coordinates

∂2Hz

∂r2+ 1

r

∂Hz

∂r+ 1

r2

∂2Hz

∂φ2+ n2k2

0Hz = 0 (5.22)

5.2.3 Solution of the wave equation in cylindrical coordinates

Here, we will analyse the equation for Ez. An equation for Hz can be solved in a similarway. Assume that the following separation of variables is possible:

Ez (r, φ) = R (r)� (φ) (5.23)

After separation, one obtains two equations

d2�(φ)

dφ2+ m2�(φ) = 0 (5.24)


0 5 10−0.5

0

0.5

1

z

J m(z

)

m = 1m = 2

m = 0

0 5 10

−4

−2

0

z

Ym

(z)

m = 0 m = 1 m = 2

Fig. 5.6 Ordinary Bessel functions. (left) Functions Jm, (right) functionsYm.

and

r2 d2R

dr2+ r

dR

dr+ (

r2q2 − m2)

R = 0 (5.25)

where m2 is a separation constant. Eq. (5.25) is known as the Bessel equation and itssolutions are known as Bessel functions [7]. The solution of Eq. (5.24) is

�(φ) = eimφ (5.26)

Here, m takes integer values because �(φ + 2π) = �(φ).The Bessel equation will be written and solved separately for core and cladding regions.

In the core region it reads

r2 d2R

dr2+ r

dR

dr+ (

r2κ2 − m2)

R = 0 (5.27)

where

κ2 = n21k2

0 − β2 > 0 (5.28)

The solutions of this equation are known as ordinary Bessel functions Jm and Nm of the firstorder and are shown in Fig. 5.6.

The Bessel equation for the cladding region is

r2 d2R

dr2+ r

dR

dr+ (

r2γ 2 + m2)

R = 0 (5.29)

where

γ 2 = β2 − n22k2

0 > 0 (5.30)

The solutions of this equation are known as modified Bessel functions, Km,Im of the secondkind and of order m. Those functions are plotted in Fig. 5.7.

MATLAB codes for obtaining plots of Bessel functions are provided in Appendix 5C.1.Using the above results for solutions of Bessel equations, one can obtain the solutions for


0 1 2 30

1

2

3

4

5

z

Km

(z)

m = 2

0 1 2 30

1

2

3

4

5

z

I m(z

)

m = 0 m = 1 m = 2

m = 0 m = 1

Fig. 5.7 Modified Bessel functions. (left) FunctionsKm, (right) functions Im.

core and cladding regions as [4]

R (r) ={

AJm (κr) + A′Ym(κr), r ≤ a

CKm (γ r) + C′Im(γ r), r > a(5.31)

where A, A′,C,C′ are constants.We expect the following properties of guided modes:

1. the solution in the core will be oscillatory and finite at r = 02. the solution in the cladding will decrease monotonically as radius increases, i.e.

field → 0 for r → ∞.

The above properties imply that for core the Jm function should be used and since Ym hasa singularity (see Fig. 5.6), for r = 0 we set A′ = 0. Similarly for cladding the function Im

increases (see Fig. 5.7) and we set C′ = 0. In cladding we use only Km.In this way one finds the solutions

Ez ={

AJm (κr) eimφ, r ≤ a

CKm (γ r) eimφ, r > a(5.32)

In exactly the same way

Hz ={

BJm (κr) eimφ, r ≤ a

DKm (γ r) eimφ, r > a(5.33)

Employing Eqs. (5.16), (5.18), (5.19) and (5.20), one can express the other field compo-nents in terms of Ez and Hz. After a little algebra, the full solutions are:

Case: r < a (core)

Eφ = − j

κ2

(jβ

rmAJm (κr) −ωμκBJ ′

m (κr)

)e jmφ (5.34)

Er = − j

κ2

(βκAJ ′

m (κr)+ jωμ

rmBJm (κr)

)e jmφ (5.35)

Hφ = − j

κ2

(ωε1κAJ ′

m (κr) + jβ

rmBJm (κr)

)e jmφ (5.36)


Hr = − j

κ2

(βκBJ ′

m (κr) − jωε1

rmAJm (κr)

)e jmφ (5.37)

Case: r > a (cladding)

Eφ = j

γ 2

(jβ

rmCKm (γ r) −ωμγ DK ′

m (γ r)

)e jmφ (5.38)

Er = j

γ 2

(βγCK ′

m (γ r) + jωμ

rmDKm (γ r)

)e jmφ (5.39)

Hφ = j

γ 2

(ωε2γCK ′

m (γ r) + jβ

rmDKm (γ r)

)e jmφ (5.40)

Hr = j

γ 2

(βγ DK ′

m (γ r) − jωε2

rmCKm (γ r)

)e jmφ (5.41)

where we have introduced the following definitions

J ′m (κr) = dJm (κr)

d(κr)

and

K ′m (γ r) = Km (γ r)

d(γ r)

5.2.4 Boundary conditions andmodal equation

We use boundary conditions derived previously which require that all field componentstangent to the core-cladding boundary at r = a must be continuous across it. We thereforehave [8]

Ez, Hz, Eφ, Hφ are continuous at r = a (5.42)

Applying the above requirement, we obtain the following equations:

1. from continuity of Ez

AJm (κa) = CKm (γ a) (5.43)

2. from continuity of Hz

BJm (κa) = DKm (γ a) (5.44)

3. from continuity of Eφ

βm

κ2aJm (κa) A + j

ωμ

κJ ′

m (κa) B + βm

γ 2aKm (γ a)C + j

ωμ

γK ′

m (γ a) D = 0 (5.45)

4. from continuity of Hφ

− jωε1

κJ ′

m (κa) A + βm

aκ2Jm (κa) B − j

ωε2

γK ′

m (γ a)C + βm

aγ 2Km (γ a) D = 0 (5.46)


The above form a set of simultaneous equations for unknown coefficients A, B,C, D. Thenontrivial solution is obtained if the determinant vanishes, i.e.∣∣∣∣∣∣∣∣∣∣∣∣

Jm (κa) 0 −Km (γ a) 00 Jm (κa) 0 −Km (γ a)

βm

κ2aJm (κa) j

ωμ

κJ ′

m (κa)βm

γ 2aKm (γ a) j

ωμ

γK ′

m (γ a)

− jωε1

κJ ′

m (κa)βm

aκ2Jm (κa) − j

ωε2

γK ′

m (γ a)βm

aγ 2Km (γ a)

∣∣∣∣∣∣∣∣∣∣∣∣= 0 (5.47)

Expansion and evaluation of this determinant is described in Appendix 5B. The resultingcharacteristic equation is[

J ′m(κa)

κJm (κa)+ K ′

m (γ a)

γ Km (γ a)

][k2

0n21J ′

m (κa)

κJm (κa)+ k2

0n22K ′

m (γ a)

γ Km (γ a)

]= β2m2

a2

(1

κ2+ 1

γ 2

)(5.48)

This equation forms the basis for obtaining propagation constants for guided modes prop-agating in cylindrical fibre. Before we start a detailed discussion, let us look at the generalclassification of modes.

5.2.5 Mode classification

Variable m is the main parameter used in mode classification. It is known as azimuthalmode number. Because of the oscillatory behavior of Jm (κa) for a given value of m asseen in Figs. 5.6, multiple solutions exist for each integer value m. We denote them as βmn,where n = 1, 2, . . . Index n is called the radial mode number and it represents the numberof radial modes that exist in the field distribution [4].

In general, electromagnetic waves in a step index optical cylindrical fibre can be dividedinto three distinct categories [9]:

1. Transverse electric (TE); sometimes designated as magnetic (H) waves. They are char-acterized by Ez = 0 and Hz �= 0.

2. Transverse magnetic (TM); sometimes designated as electric (E) waves. They are char-acterized by Ez �= 0 and Hz = 0.

3. Hybrid waves. Characterized by Ez �= 0 and Hz �= 0.

In more detail, the following cases are treated separately [8]:

1. m = 0. Still, solutions are separated into two groups:a. T M0n (transverse magnetic) because Eφ = Hr = Hz = 0. In this case coefficients

B = D = 0.b. T E0n (transverse electric) because Hφ = Er = Ez = 0. In this case coefficients

A = C = 0.2. m �= 0. Hybrid modes.

The modes are called EH or HE modes (terminology has historical roots traced tomicrowaves).


Values of β will correspond to modes which have finite components of both Ez and Hz.There are neither T E nor T M modes. General classification is [10], [11], [12].

if A = 0 the mode is called T E modeif B = 0 the mode is called T M modeif A > B = 0 the mode is called HE mode ( Ez dominates Hz)if A < B = 0 the mode is called EH mode ( Hz dominates Ez).

Now, we will discuss some of those cases in more detail.

5.2.6 Modes withm = 0

The simplest solutions of characteristic Eq. (5.48) are obtained for m = 0. In this case thereis no angular dependence on φ, and therefore field components Ez and Hz are rotationallyinvariant. The characteristic equation is reduced into two equations which describe TE andTM modes:

TE modesJ ′

0(κa)

κJ0 (κa)+ K ′

0 (γ a)

γ K0 (γ a)= 0 (5.49)

TM modesJ ′

0 (κa)

κJ0 (κa)+n2

2

n21

K ′0 (γ a)

γ K0 (γ a)= 0 (5.50)

Using the following relations for Bessel functions, (see Appendix 5A)

J ′0(u)

J 0 (u)= J−1(u)

J 0 (u)and

K′0 (w)

wK0 (w)= − K−1 (w)

wK0 (w)

and J−1 = (−1)1J1, K−1 = K1, one obtains

J1(u)

J0(u)+ u

w

K1(w)

K0(w)= 0 (5.51)

andJ1(u)

J0(u)+ n2

2

n21

u

w

K1(w)

K0(w)= 0 (5.52)

where we used definitions u = aκ and w = aγ .The interpretation of relations (5.51) and (5.52) as describing TE and TM modes comes

from the following observations.From continuity equations for Ez and Hz (established in the previous section), one can

express constants C and D in terms of A and B. Substitute these results into the continuityequation for Eφ and obtain

Aβm

a

(1

κ2+ 1

γ 2

)+ jωμ

[J ′

m(κa)

κJm (κa)+ K ′

m (γ a)

γ Km (γ a)

]B = 0

When the term in the square bracket is zero, then A = 0 and because of Eq. (5.32),Ez = 0 which means that electric field is transverse, i.e. T E mode. Similarly, substituting


expressions for constants C and D into continuity equation for Hφ one obtains

− j

[n2

1

J ′m(κa)

κJm (κa)+ n2

2

K ′m (γ a)

γ Km (γ a)

]A + βm

aωμ

(1

κ2+ 1

γ 2

)B = 0

When the term in the square bracket is zero, then B = 0 and because of Eq. (5.33), Hz = 0which means that magnetic field is transverse, i.e. T M mode.

Interpretation

Here, we will provide an interpretation of relations (5.51) and (5.52) which describe TEand TM modes.

1. TM modesAssuming B = D = 0 (and also m = 0), from Eqs. (5.34)–(5.37) in the core region one

finds

Eφ = 0, Er = − j

κβAJ ′

0 (κr)

Hφ = − j

κωε1AJ ′

0 (κr) , Hr = 0

Also, from (5.32) and (5.33) one has

Ez = AJ0 (κr) , Hz = 0

For such modes longitudinal magnetic component Hz is zero and it is therefore called TMmode (transverse magnetic). Designation of such modes is T M0n.

2. TE modesIn a similar way, taking A = C = 0 (and also m = 0), from Eqs. (5.34)–(5.37) in the

core region one obtains

Eφ = j

κωμBJ ′

0 (κr) , Er = 0

Hφ = 0, Hr = − j

κβBJ ′

0 (κr)

and also, from (5.32) and (5.33) one has

Ez = 0, Hz = BJ0 (κr)

For such modes longitudinal electric component Ez is zero and it is therefore called TEmode (transverse electric). Designation of such modes is T E0n.

A summary of mode properties for m = 0 is provided in Table 5.1.

5.2.7 Weakly guiding approximation (wga)

General solution of the dispersion Eq. (5.48) is very complicated. Gloge [13] developed anapproximation known as the weakly guiding approximation (wga). It is based on the fact


Table 5.1 Classification of modes form = 0.

Mode Values of Nonzero field Zero fielddescription Equation coefficients components components

T E0nJ ′

0(u)

uJ0 (u)+ K ′

0 (w)

γ K0 (w)= 0 A = C = 0 Hz, Hr, Eφ Ez = Er = Hφ = 0

T M0nJ ′

0(u)

uJ0 (u)+ n2

2

n21

K ′0 (w)

γ K0 (w)= 0 B = D = 0 Ez, Er, Hφ Hz, Hr, Eφ = 0

that n1 −n2 � 1. Under such approximation n21 ≈ n2

2 = n2 and also β2 ≈ k20n2. Introducing

definitions u = κa and w = γ a, general Eq. (5.48) reduces to

J ′m(u)

uJm (u)+ K ′

m (w)

wKm (w)= ±m

(1

u2+ 1

w2

)(5.53)

One can observe that two possible cases exist. We will consider them separately.

Case ‘+’:Applying properties of Bessel functions (see Appendix 5A) given by Eqs. (5.85) and (5.87),from Eq. (5.53) one obtains

Jm+1(u)

uJm (u)+ Km+1 (w)

wKm (w)= 0 (5.54)

Case ‘−’:Applying properties of Bessel functions (see Appendix 5A) given by Eqs. (5.84) and (5.86),from Eq. (5.53) one has

Jm−1(u)

uJm (u)+ Km−1 (w)

wKm (w)= 0 (5.55)

Eigenvalue Eq. (5.53) is solved numerically to find propagation constant β. In general,it has multiple solutions for each m. They are labelled by n (n = 1, 2, . . .). Thus β = βmn.Each value of βmn corresponds to one possible mode. Fibre modes are referred to as hybridmodes and denoted by [4]

HEmn (dominatesHz)

EHmn (dominatesEz)

In special case m = 0

HE0n ≡ T E0n

EH0n ≡ T M0n

}correspond to transverse electric and transverse magnetic

LPmodes

LP modes originate in a weakly guiding approximation. Those approximate solutions aremuch simpler than the exact solutions. Solving characteristic equation leads to similareigenvalues for both the EHm−1,n modes and HEm+1,n [9]. If the fields of these modes are


combined, the resulting transversal field will be linearly polarized. The linearly polarizedmodes are labelled as LPm,n. They are composed in the following way:

LP0n = HE1n

LP1n = HE2n + E0n + H0n

LPm,n = HEm+1,n + EHm−1,n, m > 2

For each n there are two LP0n modes, polarized perpendicular to each other, while for eachn and each m > 0 there are four LPm,n. Here

LPmn (LP stands for linearly polarized) (5.56)

At the end, let us summarize some important definitions:

N ≡ β

k0

which is known as mode index or effective index, and also

V = k0a(nn2

1 − n22

)1/2 ≈ 2π

λan1

√2�

where

� = n1 − n2

n1

and finally

b =β

k0− n2

n1 − n2≡ n − n2

n1 − n2

To finish this section, consider an example.

Example Typical values (for multimode fibres) are a = 25 µm, � = 5 × 10−3, V ≈ 18,at λ = 1.3 µm. Using approximate expression M = V 2/2, where M is the total number ofmodes (see Problem 1), one finds out that this particular fibre supports approximately 162modes.

5.2.8 The unified expression

Equations derived so far for TE, TM, EH and HE modes in the weakly guiding approxi-mation can be written in the form of a single equation. One should notice that in the wga,equations describing TE and TM modes degenerate into a single equation. The Eq. (5.55)for HE modes requires modification. First, write Eq. (5.55) as

uJm (u)

Jm−1(u)+ wKm (w)

Km−1 (w)= 0 (5.57)

In order to appropriately modify the above equations, consider general formulas for Besselfunctions provided in the Appendix, Eqs. (5.79) and (5.80). By changing index as m+1 → m


in those equations, one obtains

uJm(u) = 2(m − 1)Jm−1(u) − uJm−2(u)

wKm(w) = 2(m − 1)Km−1(w) + wKm−2(u)

The above equations can now be substituted into (5.57) and finally obtain

−uJm−2 (u)

Jm(u)= wKm−2 (w)

Km (w)(5.58)

Equations (5.54) and (5.58) can now be written as a single equation if we define new indexk as

k =⎧⎨⎩

1 for both TE and TM modesm + 1 for EH modesm − 1 for HE modes

The universal equation reads

uJk−1 (u)

Jk(u)= −wKk−1 (w)

Kk (w)(5.59)

5.2.9 Universal relation for fundamental modeHE11

Fundamental mode HE11 (or LP01) has the following dispersion relation (corresponding tom = 1):

J1(u)

J0(u)+ u

w

K1(w)

K0(w)= 0 (5.60)

where u = a√

n21k2

0 − β2 and w = a√

β2 − n22k2

0 . Recall previous relations

b = β2/k20 − n2

2

n21 − n2

2

= β2 − n22k2

0

n21k2

0 − n22k2

0

and

V = k0a(n2

1 − n22

)1/2

From these definitions, one finds

b = w2

V 2

After a little algebra

u2 = a2(n2

1k20 − β2

) = a2n21k2

0 − a2n22k2

0 + a2n22k2

0 − a2β2

= V 2 − w2 = V 2 − V 2b

The arguments in Eq. (5.60) can therefore be expressed as

u = V√

1 − b

and

w = V√

b


0 0.5 1 1.5 2 2.50

0.5

1

1.5

b

d(bV )/dVVd2(bV )/dV 2

V

Fig. 5.8 Plot of the waveguide parameter b and its derivatives d(bV )/dV andV d2(bV )/dV 2 as a function of theVnumber forHE11 mode.

Replacing arguments in Eq. (5.60) with the above relations, we finally obtain

J1(V√

1 − b)

J0(V√

1 − b)+

√1 − b√

b

K1(V√

b)

K0(V√

b)= 0 (5.61)

The above formula holds for mode HE11. It contains only two variables V and b. It istherefore called a universal relation. In Appendix 5C.2 we provided MATLAB programwhich plots b = b(V ) relation and also two other functions d(bV )

dV and V d2(bV )

dV 2 which areimportant in determining dispersion relation for propagating mode. Those relations areshown in Fig. 5.8.

5.2.10 Single-mode fibres

The single-mode fibre supports only HE11 mode (fundamental mode which corresponds tom = 0). From Eq. (5.48) we have

κJ0 (κa) K ′0 (γ a) + γ J ′

0 (γ a) K0 (γ a) = 0

κn22J0 (κa) K ′

0 (γ a) + γ n21J ′

0 (γ a) K0 (γ a) = 0

At cutoff, b = 0 which gives that β/κ0 = N and that γ = 0. Also, κ2a2 = a2k20 (n2

1 − n22).

Thus V = κa when γ = 0.Thus cutoff condition is

J0 (V ) = 0 (5.62)

which gives (for single mode)

V = 2.405 (5.63)

123 Dispersion

1m = 0

m = 1

0.5

0J m

(κ a

)

–0.5 κca = 0

κca = 2.405 κca = 3.85 κca = 5.52 κca = 7.02–1

–1.50 2 4

κ a6 8 10

Fig. 5.9 Plot of Bessel functions form = 0 andm = 1 used to illustrate cutoff conditions.

Example Estimates of a.Assume λ = 1.2µm, n1 = 1.45 and � = 5×10−3. For those values, the above condition

for V gives a < 3.2 µm.

5.2.11 Cutoff conditions

The condition b = 0, which results in β2 = k20n2

2, defines the so-called cutoff of the mode[14]. A mode is cut off when its field in the cladding becomes evanescent and is detachedfrom the guide. At cutoff β = k0n2 and also b = 0,V = Vc. The situation is illustrated inFig. 5.9 for m = 0 and m = 1. Case m = 0 corresponds to modes T E0n and T M0n. Thecutoff conditions for those modes are obtained from roots of the following equation:

J0(κca) = 0 (5.64)

Cutoff conditions are illustrated in Fig. 5.9. For m = 0 the few lowest roots are shown. Theycorrespond to κca = 2.405 and κca = 5.52. For m = 1 the lowest cutoffs are: κca = 3.83and κca = 7.02.

5.3 Dispersion

Dispersion is an effect responsible for spreading of a signal in time when it propagates.Signal is represented by pulse, so effectively we talk about pulse broadening.

Transmitted pulses broaden when they propagate in dispersive media. After long propa-gation it might be impossible to distinguish individual pulses, see Fig. 5.10.


(a) (b)

Fig. 5.10 The effect of dispersion on short temporal pulses. (a) Before entering dispersive medium. (b) After propagation in adispersive medium.

The main types of dispersion are [10]:

1. Material dispersion.It exists when refractive index n(λ) depends on wavelength. As a result, different wave-lengths travel with different velocities.

2. Modal dispersion.Exists in waveguides with more than one propagating mode. Each mode propagates withdifferent velocity.

3. Waveguide dispersion.Exists because propagation constant β depends on the wavelength, so even for n(λ) =const different wavelengths will propagate with different speeds. Usually the smallest,but important close to the wavelength where dispersion is zero.

5.3.1 Group delay-general discussion

In a fibre, different spectral components of the pulse travel at slightly different groupvelocities. They produce an effect known as group-velocity dispersion (GVD), also knownas intermodal dispersion or fibre dispersion. The name used depends on what descriptionis used (geometrical optics or wave optics).

Detailed discussion is based on the following assumptions:

• optical signal excites all modes equally at the input of the fibre• each mode contains all spectral components• each spectral component travels independently (in the fibre)• each spectral component undergoes a time delay or group delay.

Transit time of a pulse travelling through the fibre of length L is

τg

L= 1

vg(5.65)

where τg – time delay or group delay, L – distance, vg – group velocity, β – propagationconstant.

Group delay determines the transit time of a pulse travelling through the fibre of lengthL. Group velocity is defined as

1

vg≡ dβ

dω(5.66)

125 Dispersion

Typically, one can expresses τg in terms of derivatives with respect to k or λ or V . For thatpurpose we will summarize several relations:

ω = 2πν, ν = c

λ, k = 2π

λ, ω = kc, V = (

n21 − n2

2

)1/2 · a · k

From the above relations, one finds the following useful formulas:

d

dω= d

cdk, dk = dk

dλdλ,

dk

dλ= −2π

λ2,

d

dk= − λ2

2π

d

dλ(5.67)

To evaluate derivative with respect to V , one observes that β = β(k(V )). Therefore

dβ

dk= dβ

dV

dV

dk= V

k

dβ

dV

where we applied the following result obtained using expression for V :

dV

dk= V

k

Using the above relations allows us to express τg in several forms as

τg = Ldβ

dω= L

c

dβ

dk= L

c

V

k

dβ

dV= − Lλ2

2πc

dβ

dλ(5.68)

Because τg depends on λ, each spectral component of any particular mode travels the samedistance in different time.

At this stage, a typical procedure is to explicitly consider two main physical mechanismswhich contribute to group delay τg. One can therefore write

τg = L

c

dβ

dk

∣∣∣∣n �=const

+ L

c

dβ

dk

∣∣∣∣n=const

≡ τmat + τwg (5.69)

where τmat is known as material dispersion whereas τwg is a waveguide dispersion.Material dispersion accounts for the fact that refractive index depends on wavelength; i.e.

n = n(λ). If we neglect dependence of refractive index on wavelength, i.e. set n = const,then due to the fact that group velocity depends on λ one obtains the contribution knownas waveguide dispersion. We now consider both contributions separately.

5.3.2 Material dispersion: Sellmeier equation

Material dispersion originates from the fact that refractive index n(λ) is wavelength depen-dent. Propagation constant β is

β = 2π

λn(λ)

Use the above and Eq. (5.68) to obtain

τmat = L

c

(n − λ

dn

dλ

)(5.70)


0.5 1 1.5 21.44

1.445

1.45

1.455

1.46

1.465

wavelength (μm)

refr

activ

e in

dex

Fig. 5.11 Wavelength dependence of refractive index based on the Sellmeier equation for SiO2.

Let us concentrate now on material dispersion in silica. In practice, very often an empiricalexpression for the refractive index of glass in terms of the wavelength λ is used. It is knownas the Sellmeier equation and it was introduced in Chapter 2. Its form is

n2 = 1 + G1λ2

λ2 − λ21

+ G2λ2

λ2 − λ22

+ G3λ2

λ2 − λ23

(5.71)

Here G1, G2, G3 and λ1, λ2, λ3 are constants (called Sellmeier coefficients) which aredetermined by fitting the above expression to the experimental data. Often in literaturemore elaborate expressions are used. Using data provided in Table 2.2, we plotted inFig. 5.11a refractive index for SiO2 as a function of wavelength. MATLAB code is providedin Appendix 5C.3.

5.3.3 Waveguide dispersion

Waveguide dispersion exists because group velocity vg depends on the wavelength. Theexpression for waveguide dispersion τwg is expressed in terms of quantity b introducedearlier, so we start with the definition of parameter b

b = β2/k2 − n22

n21 − n2

2

≈ β/k − n2

n1 − n2

where we used the fact that β/k ≈ n1 ≈ n2. We use the above equation to determine β as

β = kn2(1 + b�) (5.72)

Here � = n1−n2n2

. Waveguide dispersion is obtained by evaluatind derivative dβ/dk usingEq. (5.72) and assuming that n2 is k independent. One finds

dβ

dk= n2 + n2�

d(bV )

dk

127 Pulse dispersion during propagation

Substitute the above result into general definition (5.69) and have

τwg = L

c

(n2 + n2�

d(V b)

dV

)(5.73)

5.4 Pulse dispersion during propagation

Because τg depends on λ, each spectral component of any particular mode travels the samedistance in different time. As a result, optical pulse spreads out with time. Let �λ be aspectral width of a source. Each wavelength component within �λ will propagate withdifferent group velocity, resulting in a temporal broadening of a pulse. Pulse broadening�τ is expressed as

�τ = dτ

dλ�λ (5.74)

Assuming only material dispersion, i.e. τ = τmat and using previous result (5.70) we have

dτmat

dλ= L

c

{dn

dλ− dn

dλ− λ

d2n

dλ2

}= −L

cλ

d2n

dλ2

Therefore �τmat can be written as

�τmat = −L

c

(λ2 d2n

dλ2

)(�λ

λ

)(5.75)

From the above, one can see that material dispersion can be specified in units of picosecondsper kilometre (length of fibre) per nanometre (spectral width of the source). One usuallyintroduces

Dmat = �τmat

L�λ= 1

L

dτmat

dλ=

= −1

cλ

d2n

dλ2(5.76)

which is known as material dispersion coefficient. In practical units [14]

Dmat = − 1

λ · c

(λ2 d2n

dλ2

)× 109 ps

km · nm(5.77)

where λ is in nm and velocity of light c = 3×105 km s−1. Typical behaviour for SiO2−13.5%GeO2 is shown in Fig. 5.12. MATLAB code is provided in Appendix 5C.4.

For waveguide dispersion, we can also determine waveguide dispersion coefficient Dwg

which is defined as

Dwg = 1

L

�τwg

�λ

Here �τwg is the pulse broadening due to waveguide. It can be expressed as

�τwg = dτwg

dλ�λ


1.2 1.3 1.4 1.5 1.6−20

−15

−10

−5

0

5

10

15

20

wavelength (μm)

Mat

eria

l dis

pers

ion

(ps/

km n

m)

Fig. 5.12 Material dispersion as a function of optical wavelength.

Using Eq. (5.73), one can evaluate dτwg

dλ. It is, however, typical to express it in terms of

parameter V , which is V = 2πλ

an2

√2� ≡ A

λ. Taking derivative

dV

dλ= − A

λ2

allows us to expressd

dλ= − A

λ2

d

dV

Using the above, we have

dτwg

dλ= − A

λ2

dτwg

dV= −V

λ

L

cn2�

d2(V b)

dV 2

Finally

Dwg = −n2�

c · λV

d2(V b)

dV 2(5.78)

Combining the above results for material and waveguide dispersion and using the Sellmeierequation with parameters for silica, one can determine total dispersion of fibre glass. Thework is left as a project.

5.5 Problems

1. Based on ray theory, estimate the number of modes M in a multimode fibre when Mis large.

2. Determine the cutoff wavelength for single-mode operation of an optical fibre. As-sume the following parameters: a = 5 µm, n2 = 1.450, � = 0.002.

3. Determine the value of waveguide dispersion for the above parameters.

129 Appendix 5A

4. A fibre of 10 km length has an attenuation coefficient of 0.6 dB km−1 at 1.3 µm and0.3 dB km−1 at 1.55 µm. Assuming that 10 mW of optical power is launched into thefibre at each wavelength, determine the output power at each wavelength.

5.6 Projects

1. Write MATLAB code to analyse the combined effects of material and waveguide dis-persion.

2. Using our results developed so far, establish methodologies of designing a single-modefibre with predefined properties.

3. Based on the literature, analyse polarization-maintaining fibres.

Appendix 5A: Some properties of Bessel functions

Here, we will provide several properties of Bessel functions needed in the main text. Formore information on Bessel functions consult books by Arfken [7] and Okoshi [8]. Somegeneral formulas are

Jm+1(u) + Jm−1(u) = 2m

uJm(u) (5.79)

Km+1(w) + Km−1(w) = 2m

uKm(w) (5.80)

2J ′m = Jm−1 − Jm+1 (5.81)

−2K ′m = Km−1 + Km+1 (5.82)

also

J−m = (−1)mJm, K−m = Km (5.83)

Start with (5.81) and replace Jm+1 using (5.79). One finds

J ′m(u)

uJm(u)= Jm−1(u)

uJm(u)− m

u2(5.84)

Similarly, replacing Jm−1 in Eq. (5.79) one finds

J ′m(u)

uJm(u)= m

u2− Jm+1(u)

uJm(u)(5.85)

Similar expressions can be derived for functions Km. Using (5.82) and eliminating Km+1,one has

K ′m(u)

wKm(u)= −Km−1(u)

wKm(u)− m

w2(5.86)

Finally, eliminating Km−1 in (5.82) one has

K ′m(u)

wKm(u)= −Km+1(u)

wKm(u)+ m

w2(5.87)


Appendix 5B: Characteristic determinant

In this appendix we provide details of the evaluation of characteristic determinant. Thedeterminant which describes guided modes has been obtained before and it is given byEq. (5.47):∣∣∣∣∣∣∣∣∣∣∣∣

Jm (κa) 0 −Km (γ a) 0βm

κ2aJm (κa) i

ωμ

κJ ′

m (κa)βm

κ2aKm (γ a) i

ωμ

γK ′

m (γ a)

0 Jm (κa) 0 −Km (γ a)

−iωε1

κJ ′

m (κa)βm

aκ2Jm (κa) −i

ωε2

γK ′

m (γ a)βm

aγ 2Km (γ a)

∣∣∣∣∣∣∣∣∣∣∣∣= 0

We will now evaluate it following the method of Cherin [15]. Introduce notation

A = Jm(κa), B = βm

κ2a, C = i

ω

κJ ′

m (κa)

D = Km(γ a), E = βm

aγ 2, F = i

ω

γK ′

m (γ a)

The determinant can be written as∣∣∣∣∣∣∣∣A 0 −D 00 A 0 −D

AB μC DE μF−ε1C AB −ε2F DE

∣∣∣∣∣∣∣∣ = 0

Expand the determinant. In the first step we have

A

∣∣∣∣∣∣A 0 −D

μC DE μFAB −ε2F DE

∣∣∣∣∣∣− D

∣∣∣∣∣∣0 A −D

AB μC μF−ε1C AB DE

∣∣∣∣∣∣ = 0

and in the second step

A2

∣∣∣∣ DE μF−ε2F DE

∣∣∣∣− AD

∣∣∣∣μC DEAB −ε2F

∣∣∣∣+ AD

∣∣∣∣ AB μF−ε1C DE

∣∣∣∣+ D2

∣∣∣∣ AB μC−ε1C AB

∣∣∣∣ = 0

Evaluating the above, one has

A2(D2E2 + με2F 2

)− AD (−με2CF − ABED) + AD (ABDE + με1CF )

+ D2(A2B2 + με1C

2) = 0

Rearranging terms

με2(A2F2 + ADCF

)+ με1CD (AF + CD) + 2A2D2BE + A2D2(E2 + B2

) = 0

and finally (F

D+ C

A

)(με2

F

D+ με1

C

A

)+ (E + B)2 = 0

131 Appendix 5C



5C.1.1 bessel J.m Plots Bessel function Jm

5C.1.2 bessel Y.m Plots Bessel function Ym

5C.1.3 bessel K.m Plots Bessel function Km

5C.1.4 bessel I .m Plots Bessel function Im

5C.2.1 HE11.m Driver function for universal relations5C.2.2 f unc HE11.m Function used by driverH E115C.3 sellmeier.m Plots refractive index using Sellmeier equation5C.4 disp mat.m Determines and plots material dispersion

Substitute original definitions and extracting common factors[J ′

m(κa)

κJm (κa)+ K ′

m (γ a)

γ Km (γ a)

][k2

0n21J ′

m (κa)

κJm (κa)+ k2

0n22K ′

m (γ a)

γ Km (γ a)

]= β2m2

a2

(1

κ2+ 1

γ 2

)(5.88)

Appendix 5C: MATLAB listings


Listing 5C.1.1 Program bessel J.m. Plots Bessel function Jm.

% File name: bessel_J.m

% Plot of Bessel functions of the first kind J_m(z)

clear all


z = linspace(0,10,N_max); % creation of z arguments

%

hold on

for m = [0 1 2] % m - order of Bessel function

J = BESSELJ(m,z);

h = plot(z,J);


xlabel(’z’,’FontSize’,22);

ylabel(’J_m(z)’,’FontSize’,22);

text(1.1, 0.85, ’m = 0’,’Fontsize’,18);

text(1.8, 0.65, ’m = 1’,’Fontsize’,18);

text(3, 0.55, ’m = 2’,’Fontsize’,18);


grid on

box on

%



end

pause

close all

Listing 5C.1.2 Program bessel Y.m. Plots Bessel function Ym.

% File name: bessel_Y.m

% Plot of Bessel functions of the second kind Y_m(z)

clear all


z = linspace(0.01,10,N_max); % creation of z arguments

%

hold on


Y = BESSELY(m,z);

h = plot(z,Y);



ylabel(’Y_m(z)’,’FontSize’,22);

text(2.0, 0.7, ’m = 0’,’Fontsize’,18);

text(3.5, 0.7, ’m = 1’,’Fontsize’,18);

text(5.0, 0.7, ’m = 2’,’Fontsize’,18);

axis([0 10 -5 1.5]);

grid on

box on

%



end

pause

close all

Listing 5C.1.3 Program bessel K.m. Plots Bessel function Km.

% File name: bessel_K.m

% Plot of modified Bessel functions of the second kind K_m(z)

clear all


z = linspace(0.01,3,N_max); % creation of z arguments

133 Appendix 5C

%

hold on


K = BESSELK(m,z);

h = plot(z,K);



ylabel(’K_m(z)’,’FontSize’,22);

text(0.2, 0.8, ’m = 0’,’Fontsize’,18);

text(1.0, 0.8, ’m = 1’,’Fontsize’,18);

text(1.5, 0.8, ’m = 2’,’Fontsize’,18);

axis([0 3 0 5]);

grid on

box on

%



end

pause

close all

Listing 5C.1.4 Program bessel I.m. Plots Bessel function Im.

% File name: bessel_I.m

% Plot of modified Bessel functions of the first kind I_m(z)

clear all


z = linspace(0,3,N_max); % creation of z arguments

%

hold on


Y = BESSELI(m,z);

h = plot(z,Y);



ylabel(’I_m(z)’,’FontSize’,22);

text(1.1, 1.7, ’m = 0’,’Fontsize’,18);

text(1.75, 1.7, ’m = 1’,’Fontsize’,18);

text(2.4, 1.7, ’m = 2’,’Fontsize’,18);

grid on

box on

%




end

pause

close all

Listing 5C.2.1 Program HE11.m. MATLAB code for finding a universal relation. Theprogram determines and plots the following relations: b = b(V ), d(bV )

dV ,V d2(bV )

dV 2 .

% File name: HE11.m

% Function determines and plots universal relation for mode HE_11

clear all

format long

% It works in the range of V = [0.50 2.4]

N_max = 190;

for n = 1:N_max

VV(n) = 0.50 + n*0.01;

V = VV(n);

b_c(n) = fzero(@(b) func_HE11(b,V),[0.0000001 0.8]);

end

%

hold on

h1 = plot(VV,b_c); % plots b =b(V)

%

temp1 = VV.*b_c;

dy1 = diff(temp1)./diff(VV);

Vnew1 = VV(1:length(VV)-1);

h2 = plot(Vnew1,dy1); % plots first derivative

%

temp2 = dy1;

dy2 = diff(temp2)./diff(Vnew1);

Vnew2 = Vnew1(1:length(Vnew1)-1);

h3 = plot(Vnew2,Vnew2.*dy2); % plots second derivative

%

text(2.3, 0.6, ’b’,’Fontsize’,16);

text(1.8, 1.2, ’{d(bV)}/{dV}’,’Fontsize’,16);

text(0.25, 1.3, ’V{d^{2}(bV)}/{dV^{2}}’,’Fontsize’,16);

axis([0 2.5 0 1.5]);


xlabel(’V’,’FontSize’,22);

grid on

box on

set(h1,’LineWidth’,1.5); % new thickness of plotting lines

set(h2,’LineWidth’,1.5);

set(h3,’LineWidth’,1.5);


135 Appendix 5C

pause

close all

Listing 5C.2.2 Program func HE11.m. MATLAB function for finding a universal rela-tion.

function result = func_HE11(b,V)

% Function name: func_HE11.m

% Function which defines basic relation

u= V*sqrt(1-b);

J1 = BESSELJ(1,u);

J0 = BESSELJ(0,u);

%

w = V*sqrt(b);

K1 = BESSELK(1,w);

K0 = BESSELK(0,w);

lhs = J1./J0;

rhs = (w./u).*(K1./K0);

result = lhs - rhs;

Listing 5C.3 Program sellmeier.m. Plots refractive index for SiO2 using the Sellmeierequation.

% File name: sellmeier.m

% Plot of refractive index based on Sellmeier equation

% for Si O_2

clear all


lambda = linspace(0.5,1.8,N_max); % creation of lambda arguments

% between 0.5 and 1.8 microns

%

% Data for SiO_2

G_1 = 0.696749; G_2 = 0.408218; G_3 = 0.890815;

lambda_1=0.0690606; lambda_2=0.115662; lambda_3=9.900559; % in microns

%

term1 = (G_1*lambda.^2)./(lambda.^2 - lambda_1^2);



ref_index_sq = 1.0 + term1 + term2 + term3;

ref_index = sqrt(ref_index_sq);

%

h = plot(lambda,ref_index,’LineWidth’,1.5);



xlabel(’wavelength (\mum)’,’FontSize’,14);

ylabel(’refractive index’,’FontSize’,14);


pause

close all

Listing 5C.4 Program disp mat.m. Determines coefficient Dmat material dispersion.Refractive index is determined using the Sellmeier equation.

% File name: disp_mat.m

% Plots D_mat

% Material dispersion is determined using Sellmeier equation

% describing refractive index for pure and doped silica

% Data from Kasap2001, p.45 for SiO_2 - GeO_2

clear all

c_light = 3d5; % velocity of light, km/s

disp_min = -20; disp_max =20;


lambda_min = 1.200; lambda_max = 1.600; % in microns

% creation of lambda arguments between 1.2 and 1.6 microns

lambda = linspace(lambda_min,lambda_max,N_max);

G_1 = 0.711040; G_2 = 0.451885; G_3 = 0.704048;

lambda_1 = 0.0642700; lambda_2 = 0.129408; lambda_3 = 9.425478;

%




%

ref_index_sq = 1.0 + term1 + term2 + term3;

ref_index = sqrt(ref_index_sq);

%

%---------- Determination of material dispersion D_mat -------------

ttt1 = ref_index;

dy1_lam = diff(ttt1)./diff(lambda); % first derivative

lambda1 = lambda(1:length(lambda)-1);

ttt2 = dy1_lam;

dy2_lam = diff(ttt2)./diff(lambda1);

lambda2 = lambda1(1:length(lambda1)-1);

D_mat = (-lambda2.*dy2_lam/c_light)*1d9;

plot(lambda2,D_mat,’LineWidth’,1.5);

xlabel(’wavelength (\mum)’,’FontSize’,14);

ylabel(’Material dispersion (ps/km nm)’,’FontSize’,14);


axis([lambda_min, lambda_max, disp_min, disp_max])

137 References

grid on

pause

close all

References

[1] G. Keiser. Optical Fiber Communications. Third Edition. McGraw-Hill, Boston, 2000.[2] F. L. Pedrotti and L. S. Pedrotti. Introduction to Optics. Third Edition. Prentice Hall,

Upper Saddle River, NJ, 2007.[3] T. Miya, Y. Terunuma, T. Hosaka, and T. Miyashita. Ultimate low-loss single-mode

fibre at 1.55 µm. Electronics Letters, 15:106–8, 1979.[4] J. A. Buck. Fundamentals of Optical Fibers. Second Edition. Wiley-Interscience,

Hoboken, NJ, 2004.[5] J. C. Palais. Fiber Optic Communications. Fifth Edition. Prentice Hall, Upper Saddle

River, NJ, 2005.[6] D. K. Cheng. Fundamentals of Engineering Electromagnetics. Addison-Wesley, Read-

ing, MA, 1993.[7] G. Arfken. Mathematical Methods for Physicists. Academic Press, Orlando, FL, 1985.[8] T. Okoshi. Optical Fibers. Academic Press, New York, 1982.[9] W. van Etten and J. van der Plaats. Fundamentals of Optical Fiber Communications.

Prentice Hall, New York, 1991.[10] C. R. Pollock and M. Lipson. Integrated Photonics. Kluwer Academic Publishers,

Boston, 2003.[11] A. W. Snyder and J. D. Love. Optical Waveguide Theory. Kluwer Academic Publishers,

1983.[12] D. Marcuse. Theory of Dielectric Optical Waveguides. Second Edition. Academic

Press, New York, 1991.[13] D. Gloge. Weakly guiding fibers. Appl. Opt., 10:2252–8, 1971.[14] G. Ghatak and K. Thyagarajan. Introduction to Fiber Optics. Cambridge University

Press, Cambridge, 1998.[15] A. H. Cherin. An Introduction to Optical Fibers. McGraw-Hill, New York, 1983.

6 Propagation of linear pulses

In this chapter we outline basic issues associated with the description of linear pulses.Basic principles of two fundamental modulation formats, namely return-to-zero (RZ) andnon-return-to-zero (NRZ) modulations, will also be explained.

Formats of data transmission are one of the critical factors in optical communicationsystems. We therefore have implemented several waveforms corresponding to basic mod-ulation formats and various profiles of individual bits. Later on, some of this informationwill be used to analyse performance of communication systems. We start with a summaryof the basic pulses.

6.1 Basic pulses

In this section we will review properties of the basic pulses. Basic approximations andproperties will be described. We start with rectangular pulses.

6.1.1 Rectangular pulses

The simplest pulse, namely the rectangular one, is shown in Fig. 6.1. It is described as

srect (t) ={

s0, 0 < t < T0, otherwise

}(6.1)

where T is the pulse duration and s0 its magnitude. It can be represented as a Fourier series

srect (t) =∞∑

n=1

sn sin(nπ

Tt)

(6.2)

where

sn ={

0, n even4s0πn , n odd

(6.3)

In Fig. 6.1 we plotted three possible approximations to a rectangular pulse for n = 1,

n = 3, n = 5 corresponding to different number of terms in Eq. (6.2). One can observe thatwith the increasing number of terms in Fourier expansion more accurate approximationto a rectangular pulse is possible. MATLAB code used to create Fig. 6.1 is shown in theAppendix, Listing 6A.1.

138

139 Basic pulses

0 2 4 6 8 10 12−0.5

0

0.5

1

1.5

time (a.u.)

s (a.

u.)

n = 1 n = 3 n = 5

Fig. 6.1 Rectangular pulse of height 1 (a.u.) approximated by varying number of terms in Fourier series expansion.

In practice, one often considers a symmetrical rectangular pulse shown in Fig. 6.3. It isdefined as

srect,symm(t) ={

1, |t| < τ

0, |t| > τ(6.4)

Its Fourier transform can be evaluated analytically as follows:

Srect,symm(ω) =∫ +∞

−∞x(t) e−iωt dt =

∫ +τ

−τ

e−iωt dt = 1

−iωe−iωt

∣∣∣∣+τ

−τ

= 1

−iω

(e−iωτ − eiωτ

) = 2 sin ωτ

ω(6.5)

where we have used Euler identity. A plot of Eq. (6.5) is shown in Fig. 6.2.To sum up the discussion of rectangular pulses, we have evaluated Fourier transform of

a rectangular pulse using MATLAB built-in functions. A plot of a symmetric rectangularpulse and its MATLAB generated absolute value of Fourier transform are shown in Fig. 6.3.Plots were created by MATLAB code shown in Appendix, Listing 6A.2.

6.1.2 Gaussian pulses

A Gaussian pulse is considered an accurate model of the data waveforms generated inpractical optical communication systems. The Gaussian profile is described as

sG(t) = A

σ√

2πe− 1

2 (tσ )

2

(6.6)

The pulse has an area A = P0τ0, where τ = σ√

2π and P0 is its maximum value (at t = 0).Its spectral function SG(ω) is obtained by Fourier transform and it is also a Gaussian


−2 −1 0 1 2−4

−2

0

2

4

6

8

10

frequency (a.u.)

Four

ier

tran

sfor

m (

S)

(a.u

.)A B

Fig. 6.2 Fourier transform of rectangular pulse of duration τ and areaA. Point A corresponds to the frequency ν = 2/τ andpoint B to frequency ν = 3/τ .

−20 −10 0 10 200

0.5

1

1.5

2

time (a.u.)

s r (a.

u.)

−40 −20 0 20 40−0.5

0

0.5

1

1.5

2

2.5

3 × 104

Four

ier t

rans

form

of r

ecta

ngul

ar p

ulse

(a.u

.)

Fig. 6.3 Rectangular pulse (left) and the absolute value of its Fourier transform determined using MATLAB functions.

function, see Fig. 6.4:

SG(ω) = Ae− 12 (ωσ )2

(6.7)

6.1.3 Super-Gaussian pulse

A super-Gaussian pulse is a generalization of the usual Gaussian and is

sSG(t) = A

σ√

2πe− 1

2 (tσ )

2m

(6.8)

141 Basic pulses

0 t

sG(t)

Pτ τ

τ τ

0

0

0

0P0

SG(ω)

0

10

12 (=B )0

ω

Fig. 6.4 Schematic illustration of Gaussian pulse in time and its Fourier transform.

0 50 100 150 200 250 3000

0.002

0.004

0.006

0.008

0.01

0.012

0.014

time

ampl

itud

e

Fig. 6.5 Super-Gaussian pulses for several values ofm = 1, 2, 3, 4.

The parameter m controls the degree of edge sharpness. For m = 1 one recovers the ordinaryGaussian pulse. For larger values of m the pulse becomes closer to a rectangular pulse, seeFig. 6.5. (Figure generated by MATLAB code, in Appendix, Listing 6A.3.)

6.1.4 Chirped Gaussian pulse

Chirp refers to a process which changes the frequency of a pulse with time. Chirp pulses aremost commonly created during direct modulation of a semiconductor laser diode (semicon-ductor lasers are discussed in Chapter 7). Mathematical expression for a chirped Gaussianpulse at z = 0 is [1], [2]

s(z = 0, t) = Re

{exp

[−1 + iC

2

(t

T0

)2]

exp (−iω0t)

}

= e− 1

2

(t

T0

)2

cos

[ω0t + 1

2C

(t

T0

)2]

(6.9)


0 10 20 30 40 500

0.5

1Gaussian pulse

time

ampl

itud

e

0 10 20 30 40 50−1

0

1Chirped Gaussian pulse

time

ampl

itud

e

Fig. 6.6 Comparison of Gaussian pulse (upper plot) and chirped Gaussian pulse (lower plot).

where T0 determines the width of the pulse, C is the chirp factor which determines thedegree of chirp of the pulse and ω0 is the carrier frequency of the pulse. s(0, t) is thedimensionless function describing pulse.

Phase of such pulses is

φ(ω) = ω0t + 1

2C

(t

T0

)2

(6.10)

Change in frequency δω(t) is therefore

δω(t) = −∂φ(ω)

∂t= ω0 + C

t

T 20

(6.11)

Fourier spectrum of a chirped pulse is broader compared to unchirped one. Evaluation ofthe Fourier transform of (6.9) gives

s(0, ω) =(

2πT 20

1 + iC

)1/2

exp

[− ω2T 2

0

2 (1 + iC)

](6.12)

The spectral half-width is defined as

|s(0, ωe)|2|s(0, 0)|2 = e−1 (6.13)

For a symmetric pulse centered at t = 0, one has �ω = ωe. Detailed calculations forspectral half-width (point where the intensity drops 1/e) give [3] (see Problem 2)

�ω = (1 + C2

)T −1

0 (6.14)

Chirped Gaussian pulse is plotted in Fig. 6.6 (lower plot). For comparison, regularGaussian pulse for the same parameters is also shown (upper plot). The plots were obtainedusing MATLAB code provided in Appendix, Listing 6A.4.

143 Modulation of a semiconductor laser

DC current (mA)Li

ght o

utpu

t (m

W/fa

cet)

0 50 100 150 2000

5

10

10 Cο 50 Cο90 Cο

Fig. 6.7 Schematic illustration of light-current laser characteristics at various temperatures.

6.2 Modulation of a semiconductor laser

In practical applications optical pulses are created by semiconductor laser. Typical light-current (L-I) characteristics of the semiconductor laser are shown in Fig. 6.7. We show herean output of power-current characteristics (L-I) for several temperatures. One can observea flat region to the left until the so-called threshold and then sudden increase in power.Threshold current Ith is the forward injection current at which optical gain in the lasercavity is equal to losses. Above the threshold, stimulated emission is the dominant processin converting current to light; below the threshold is the light emission which is causedby spontaneous processes with characteristics (speed, spectrum and efficiency) similar tothose from light emitting diodes (LEDs).

In Fig. 6.8 we illustrate principles of direct modulation of a semiconductor laser. It is asimple and inexpensive process since no other components are required. The disadvantageof direct modulation is that the resulting pulses are considerably chirped.

6.2.1 Modulation formats

Various modulation formats are often discussed in literature. For detailed analysis consultbooks by Keiser [4], Liu [5], Binh [6], Ramaswami or Sivarajan [7]. Here, we discuss brieflytwo important cases only as illustrated in Figs. 6.9 and 6.10.

At a digital format light pulse intensity (say) larger than zero, represents logical ONE andits absence, logical ZERO. In the return-to-zero (RZ) format, signal drops to zero betweenthe pulses, see Fig. 6.10. Each bit has allocated time T . In RZ, pulses occupy half of thetime slot reserved for each bit, see Fig. 6.10. Also, power spectrum is shown. Since most ofthe signal power is at the lower frequency, the RZ signal is transmitted by a system havinga bandwidth of 2

T Hz.In the non-return-to-zero (NRZ) format, the pulses occupy entire time slot reserved for

each bit. NRZ pulses are twice as long as the RZ pulses. In NRZ modulation format the


0

Ligh

t out

put ON

OFF

TimeBias

Drive

Fig. 6.8 Illustration of the principle of modulation in a semiconductor laser.

time

time

RZ

NRZ

T

0 1 0 1 0

Fig. 6.9 Comparison of RZ and NRZ modulation formats. Digital bit stream is: 01010.

intensity of light remains at its high value whenever a number of consecutive ONEs aretransmitted. The bandwidth is 1/T , only half of that required for RZ system.

For encoding, practical current pulse driving the semiconductor laser is assumed to beof the form [8]

I(t) =

⎧⎪⎨⎪⎩0 t < 0

Im [1 − exp (−t/tr)] 0 ≤ t ≤ T ′

Im exp (−(t − T )/tr) T ′ < t

(6.15)

where Im is the peak modulation, tr determines the pulse rise time, and T ′ = T for NRZencoding and T ′ = T/2 for RZ coding. The bias current Ibias is typically 1.1 times the laserthreshold current.

145 Modulation of a semiconductor laser

fT

1/T 2/T

RZ

fT

1/T 2/T

NRZ

Fig. 6.10 Comparison of modulation formats (RZ and NRZ) (schematically) and their corresponding bandwidths.

0 1 2 3 4 5 6× 10−9

0

0.5

1

time

Individual pulses

0 1 2 3 4 5 6× 10−9

0

1

2

time

Total signal

Fig. 6.11 Train of Gaussian pulses for logical combination of bits 010110. Individual pulses are shown (top) and also completesignal (bottom).

6.2.2 Creation of waveforms

In practice, to evaluate photonic systems for some modulation formats one needs to generatewaveforms consisting of pulses of various shapes.

The generated sequence of Gaussian pulses (waveform) for some particular logical com-bination is illustrated in Fig. 6.11. MATLAB code is provided in Appendix, Listings 6A.5and 6A.5.1.


0 2 4 6 8× 10−9

2

2.5

3

3.5

4

4.5

5

5.5

time [s]

Cur

rent

[mA

]

Fig. 6.12 Eight-bit pattern generated in NRZ format.

Another wavefront which is used to represent driving current of semiconductor lasers inNRZ (non-return-to-zero) format can be expressed as [9]

Jmod (t) =

⎧⎪⎪⎨⎪⎪⎩Ibias + Im

(1 − e−2.2t/τr

)current bit 1, previous 0

Ibias + Ime−2.2t/τr current bit 0, previous 1Ibias current bit 0, previous 0Ibias + Im current bit 1, previous 1

(6.16)

Implementation of the above modulation format is shown in Appendix, Listing 6A.6 and atypical result in Fig. 6.12.

6.3 Simple derivation of the pulse propagation equation in thepresence of dispersion

Assume the existence of a propagating electric field E(r, t) at point r and time t. It can beFourier decomposed as

E(r, t) = 1

2π

∫ ∞

−∞E(r, ω) e− jωt dω (6.17)

Therefore electric field can be interpreted as an infinite sum of monochromatic waves, eachhaving an angular frequency ω. The inverse Fourier transform is

E(r, ω) =∫ ∞

−∞E(r, t) e jωt dt (6.18)

147 Simple derivation of the pulse propagation equation in the presence of dispersion

We further assume that Fourier component E(r, ω) can be expressed as

E(r, ω) = x B(0, ω) F (x, y) e jβz (6.19)

where unit vector x represents the state of polarization of propagating field, B(0, ω) is theinitial amplitude of the field propagating along z-direction, F (x, y) describes the transversedistribution of the field and β is the propagation constant. We made a similar assumptionearlier when we discussed propagation of the EM field in planar waveguides.

For the field propagating in an optical fibre, the transverse distribution F (x, y) is oftenassumed to be Gaussian, namely

F (x, y) = exp

(−π

x2 + y2

Ae f f

)(6.20)

where Ae f f is the effective area of the fibre.The effect of dispersion results in the frequency dependence of the propagation constant

β; i.e. β = β(ω). As a result, each frequency component in Eq. (6.17) will propagate witha different propagation constant.

To derive propagation equation for each frequency component, write Eq. (6.19) as

E(r, ω) = E(x, y, z = 0, ω) ejβz (6.21)

From above, by direct differentiation with respect to z, one finds

∂E(r, ω)

∂z= jβ(ω)E(r, ω) (6.22)

The above equation describes propagation of EM field with the dispersion induced, broad-ening included. More rigorous derivation of the above equation, starting from Maxwell’sequations, will be performed in the next section.

Propagation constant β(ω) in terms of the frequency dependent mode index n(ω) is [3]

β(ω) = n(ω)ω

c(6.23)

It is customary to expand β(ω) around the centre frequency of the pulse ω0 in a Taylorseries

β(ω) = β0 + β1 (ω − ω0) + 1

2β2 (ω − ω0)

2 + 1

6β3 (ω − ω0)

3 (6.24)

where βi = ∂ iβ

∂ωi

∣∣∣ω=ω0

. Expansion up to the third order is sufficient to describe typical effects

existing in optical fibre. More practical expressions for expansion coefficients are [3]

β1 = 1

c

[n(ω) + ω

dn(ω)

dω

]= ng

c= 1

vg(6.25)

where ng is the group index and vg group velocity. For coefficient β2 one finds

β2 =∂2β(ω)

∂ω2= ∂β1

∂ω= 1

c

[2

dn(ω)

dω+ ω

d2n(ω)

dω2

](6.26)


Coefficient β2 is related to dispersion parameter D as

D = 1

L

dτ

dλ= −2πc

λ2β2 (6.27)

Here τ is the group delay through a fibre of length L, and λ is the wavelength. Units ofparameter D are [ ps

nm·km ].To account for parameter β3 in practice, the so-called dispersion slope S is introduced:

S = dD

dλ(6.28)

Parameter S describe the variation of D with the wavelength. Units of it are [ psnm2·km ].

6.4 Mathematical theory of linear pulses

Here, we will outline a full derivation of the equation describing linear pulses based onMaxwell equations. This more formal derivation illustrates needed approximations.

Maxwell’s equations were introduced in Chapter 3; for optical fibre where there is noflow of current and no free charges these are

∇ × E = −∂B

∂t, ∇ × H = ∂D

∂t(6.29)

and

∇ · D = 0, ∇ · B = 0 (6.30)

Constitutive relations are expressed as

D = ε0E + P, B = μ0H + M (6.31)

Here P is polarization which is divided into linear and nonlinear parts:

P(r, t) = PL(r, t) + PNL(r, t) (6.32)

For a nonmagnetic medium such as fibre, the magnetization M = 0. In this chapter wediscuss only linear effects.

Linear polarization is related to electric field as

PL(r, t) = ε0

∫ ∞

−∞χ(1)(t − t ′) E(r, t ′) dt ′ (6.33)

Fourier transform and its inverse are introduced as

E(r, ω) =∫ ∞

−∞E(r, t) eiωtdt (6.34)

E(r, t) = 1

2π

∫ ∞

−∞E(r, ω) eiωt dω (6.35)

149 Mathematical theory of linear pulses

Fourier transformation of linear polarization, Eq. (6.33) gives

PL(r, ω) =∫ ∞

−∞PL(r, t) eiωtdt

= ε0

∫ ∞

−∞

∫ ∞

−∞χ(1)(t − t ′) E(r, t ′) eiωt dt ′dt

= ε0

∫ ∞

−∞

∫ ∞

−∞χ(1)(t ′′) E(r, t ′) eiωt ′′eiωt ′ dt ′dt ′′

= ε0

∫ ∞

−∞dt ′E(r, t ′) eiωt ′

∫ ∞

−∞dt ′′χ(1)(t ′′) eiωt ′′

= ε0 χ(1)(ω) E(r, ω) (6.36)

where we have made change of variables t − t ′ = t ′′. χ(1)(ω) is the Fourier transform ofthe susceptibility χ(1)(t)

χ (1)(ω) =∫ ∞

−∞dtχ(1)(t) eiωt (6.37)

To derive wave equation take ∇ × . . . operation of the first Maxwell’s equation and use thesecond Maxwell’s equation and constitutive relations. In a few steps, one obtains

∇ × ∇ × E(r, t) = − ∂

∂t∇ × B(r, t)

= − ∂

∂tμ0∇ × H(r, t)

= −μ0∂2D(r, t)

∂t2

= −μ0ε0∂2E(r, t)

∂t2− μ0

∂2P(r, t)

∂t2

In the linear regime

∇ × ∇ × E(r, t) = −μ0ε0∂2E(r, t)

∂t2− μ0

∂2PL(r, t)

∂t2

Using mathematical identity

∇ × ∇ × E = ∇ (∇ · E) − ∇2E

and third Maxwell’s equation (∇ · D = ε∇ · E = 0), one finally obtains

∇2E(r, t) + 1

c2

∂2E(r, t)

∂t2+ μ0

∂2PL(r, t)

∂t2= 0 (6.38)

The above is a linear equation in field E(r, t). Taking Fourier transform gives

∇2E(r, ω) + ω2

c2E(r, ω) + μ0ω

2PL(r, ω) = 0 (6.39)

Finally, substituting the result in (6.36) for linear polarization gives the wave equation inthe frequency domain

∇2E(r, ω) + ε(ω)ω2

c2E(r, ω) = 0 (6.40)

The above three-dimensional wave equation will be developed in one dimension next.


6.4.1 One-dimensional approach

A one-dimensional wave equation (in time domain) for a linear medium with zero dispersionis

∂2E(z, t)

∂z2− μ0ε0n2

0

∂2E(z, t)

∂t2= 0 (6.41)

where n0 is refractive index which in the absence of dispersion is frequency independent.It is related to propagation constant β0 as

β0 = n0k0 (6.42)

The wave equation describes pulses with central frequency ω0. The following relation holds:

k0 = ω0√

μ0ε0 (6.43)

Solution of the wave equation which describes linear pulse is

E(z, t) = A(z, t) ei(ω0t−β0z) (6.44)

where A(z, t) is the pulse envelope. In the following we will work within slowly varyingenvelope approximation (SVEA), which is based on the following approximations:∣∣∣∣∂2A(z, t)

∂z2

∣∣∣∣ �∣∣∣∣β0

∂A(z, t)

∂z

∣∣∣∣and ∣∣∣∣∂2A(z, t)

∂t2

∣∣∣∣ �∣∣∣∣ω0

∂A(z, t)

∂t

∣∣∣∣One needs to evaluate derivatives appearing in Eq. (6.41) using solution given by Eq. (6.44)and then apply SVEA. The resulting linear wave equation describes pulses propagatingwithout dispersion and it is

∂A(z, t)

∂z+ β0

ω0

∂A(z, t)

∂t= 0 (6.45)

In the presence of dispersion it is convenient to represent pulse in terms of Fourier compo-nents. Each such component travels with different velocity.

Taking Fourier transformation of the original pulse described by Eq. (6.44) gives

E(z, t) = 1

2π

∫ ∞

−∞E(z, ω) eiωtdω (6.46)

E(z, ω) describes propagation of spectral component with angular frequency ω inside thefibre as follows [1]:

E(z, ω) = E(0, ω) e−iβz (6.47)

Combining Eqs. (6.44), (6.34) and (6.47) gives

A(z, t) ei(ω0t−β0z) = 1

2π

∫ ∞

−∞E(0, ω) e−iβz eiωtdω

151 Mathematical theory of linear pulses

From the above, the expression for pulse envelope A(z, t) is

A(z, t) = 1

2π

∫ ∞

−∞E(0, ω) ei(ω−ω0 )t e−i(β−β0 ) zdω (6.48)

Next, let us introduce �ω = ω − ω0 and assume �ω � ω0.Pulse broadening results from frequency dependence of the propagation constant β; i.e.

β = β(ω). Because of �ω � ω0, it is customary to expand β around frequency ω0 asfollows [the same equation as (6.24)]:

β(ω) = β0 + β1�ω + 1

2β2(�ω)2 + 1

6β3(�ω)3 + . . . (6.49)

where higher order terms are neglected. The following notation has been introduced:

β0 = β(ω0), β1 = dβ

dω

∣∣∣ω=ω0

, β2 = d2β

dω2

∣∣∣ω=ω0

, β3 = d3β

dω3

∣∣∣ω=ω0

. Substituting the above

expansion into Eq. (6.48) gives

A(z, t) = 1

2π

∫ ∞

−∞E(0, ω) ei�ωt e−i�ωβ1ze−i(�ω)2 1

2 β2ze−i(�ω)3 16 β3zdω

The above expression will be used to evaluate derivatives ∂A(z,t)∂z and ∂A(z,t)

∂t . One finds

∂A(z, t)

∂t= 1

2π

∫ ∞

−∞E(0, ω) i�ω ei�ωt e−i�ωβ1ze−i(�ω)2 1

2 β2ze−i(�ω)3 16 β3zdω (6.50)

and

∂A(z, t)

∂z= 1

2π

∫ ∞

−∞E(0, ω)(−i�ωβ1) ei�ωt e−i�ωβ1ze−i(�ω)2 1


+ 1

2π

∫ ∞

−∞E(0, ω)

[−i

1

2(�ω)2β2

]ei�ωt e−i�ωβ1ze−i(�ω)2 1


+ 1

2π

∫ ∞

−∞E(0, ω)

[−i

1

6(�ω)3β3

]× ei�ωt e−i�ωβ1ze−i(�ω)2 1

2 β2ze−i(�ω)3 16 β3zdω (6.51)

Terms on the right hand side of the above equation can be expressed in terms of the timederivatives as follows:

1

2π

∫ ∞

−∞E(0, ω)(−i�ωβ1) ei�ωt e−i�ωβ1ze−i(�ω)2 1

2 β2ze−i(�ω)3 16 β3zdω = −β1

∂A(z, t)

∂t

1

2π

∫ ∞

−∞E(0, ω)

[−i

1

2(�ω)2β2



= i1

2β2

∂2A(z, t)

∂t2

1

2π

∫ ∞

−∞E(0, ω)

[−i

1

6(�ω)3β3



+ 1

6β3

∂3A(z, t)

∂t3


Using the above replacements in Eq. (6.51) finally gives

∂A(z, t)

∂z= −β1

∂A(z, t)

∂t− i

1

2β2

∂2A(z, t)

∂t2+ 1

6β3

∂3A(z, t)

∂t3(6.52)

This is the basic equation which describes pulse evolution in the linear regime in thepresence of dispersion. Its generalization to include the nonlinear effect (Kerr effect) whichdescribes optical solitons will be discussed in Chapter 15.

In the absence of dispersion, i.e. when β2 = β3 = 0, the optical pulse propagates withoutchanging its shape. In such case pulse envelope changes as A(z, t) = A(0, t − β1z).

The general equation describing pulse propagation can be simplified by going to areference frame moving with the pulse. The transformation to the new coordinates is

t ′ = t − β1z) and z′ = z (6.53)

In the reference frame moving with the pulse, Eq. (6.52) takes the form

∂A(z′, t ′)∂z′ + i

2β2

∂2A(z′, t ′)∂t ′2

− 1

6β3

∂3A(z′, t ′)∂t ′3

(6.54)

The above equation found many applications which have been discussed extensively, see[3], [1], [9].

6.5 Propagation of pulses

6.5.1 Analytical description of the propagation of a chirp Gaussian pulse

Pulse behaviour is described by the Eq. (6.54) derived previously. For β3 = 0 the equationis

∂A

∂z′ + i

2β2

∂2A

∂t ′2= 0

and pulse evolution can be determined analytically. The solution is

A(z, t) = 1

2π

∫ +∞

−∞A(0, ω) e

i2 β2ω

2z−iωtdω (6.55)

In the solution, the initial Gaussian pulse A(0, ω) is given by Eq. (6.12). SubstitutingEq. (6.12) into (6.55) and evaluating the Gaussian integral gives

A(z, t) = T0[T 2

0 − iβ2z (1 + iC)]1/2

exp

{− (1 + iC) t2

T 20 − iβ2z (1 + iC)

}(6.56)

The above result defines new pulse width T1 for a propagating pulse at position z. Its valueis

T1

T0=[(

1 + Cβ2 z

T 20

)2

+(

β2 z

T 20

)2]1/2

(6.57)

Evolution of chirped and unchirped pulses is shown in Fig. 6.13. Consult listing 6A.7for MATLAB code.

153 Propagation of pulses

0 0.5 1 1.50

0.5

1

1.5

2

2.5

3

T 1/T0

z/LD

C = −1

C = 0

C = 1C = 2

Fig. 6.13 Evolution of pulse width as a function of normalized distance z/LD for chirped and unchirped pulses. Dispersianlength is defined by Eq. (15.14).

6.5.2 Numerical method using Fourier transform

We finish this chapter by outlining the numerical method of using Eq. (6.54) to describepropagation of a Gaussian pulse (and also a super-Gaussian pulse) in the presence ofdispersion. The method consists of the following steps.

1. The single pulse p = exp(−t2/T0) is used with sampling period of T = 0.08s in thetime domain. (If needed it can be plotted using plot(t, p) MATLAB function.)

2. The Fourier transform of this single pulse is found using P = f f tshi f t( f f t(p)) codewhere p is in time domain and P is in the frequency domain.

3. The frequency domain pulse is sampled using sampling frequency =1/64. The solutionin frequency domain is found by multiplying Fourier transform of single pulse and transferfunction H (ω) where spectral function is

H (ω) = exp[(

−α

2− i

2· β2 · ω2 − i

6· β3 · ω3

)· L]

(6.58)

where α are losses, L is the propagation length and β2 and β3 are previously defineddispersion parameters.

4. The solution in time domain is obtained by using inverse Fourier transformi f f t( f f tshi f t(P prop)) and it is plotted. Consult Listings 6A.8 and 6A.8.1 for MAT-LAB code.

The propagation of Gaussian pulse over a distance of 4000 km is shown in Fig. 6.14 forthe following parameters: α = 0, β2 = 60 ps2/km and β3 = 0.01 ps3/km. MATLAB codeis provided in Appendix, Listings 6A.8.1 and 6A.8.2.


−4 −2 0 2 40

0.2

0.4

0.6

0.8

1

time

Arb

itra

ry u

nits

original pulsetransmited pulse

Fig. 6.14 Dispersion induced broadening of a Gaussian pulse.β2 = 60 ps2/km andβ3 = 0.01 ps3/km.

−4 −2 0 2 40

0.2

0.4

0.6

0.8

1

1.2

1.4

time

Arb

itra

ry u

nits

original pulsetransmited pulse

Fig. 6.15 Evolution of a super-Gaussian pulse withm = 3.β2 = 0 andβ3 = 0.01 ps3/km.

In Fig. 6.15 we show propagation of a super-Gaussian pulse with m = 3 over the samedistance with β2 = 0. The remaining parameters remain unchanged.

6.5.3 Fourier transform split-step method

This method will be extensively discussed in later Chapter 12 describing beam propagationmethod (BPM). Here we use it to illustrate propagation of Gaussian pulse. First, we brieflydescribe the method.

155 Problems

Table 6.1 Values of parameters.

Name Symbol Value

Second-order dispersion β2 60 ps/nm−1 km−1

Third-order dispersion β3 0.01 ps/nm−1 km−1

Losses α 0Width of Gaussian pulse w0 20 ps

We use the linear Schroedinger equation derived previously, Eq. (6.54). Taking Fouriertransform results in

∂A(z, ω)

∂z= − iβ2

2(−iω)2 A(z, ω) + β3

6(−iω)3 A(z, ω) − α

2A(z, ω) (6.59)

where α describes losses. Fourier transform is defined in a usual way as

A(z, ω) =∫ +∞

−∞A(z, t) eiωt dt

Eq. (6.59) can be expressed as

∂A(z, ω)

∂z= P · A(z, ω) (6.60)

where the propagator operator P is defined as

P = iβ2

2ω2 + iβ3

6ω3 − α

2(6.61)

Eq. (6.60) can be integrated over small distance h along the z-axis with the result

A(z + h, ω) = eP·h A(z, ω) (6.62)

The above equation is directly implemented in MATLAB. In Table 6.1 we summarizedvalues used in simulations.

The results are shown in Figs. 6.16 and 6.17. We show evolution of the Gaussian pulseand also a three-dimensional view of the evolving pulse. MATLAB code is provided inAppendix, Listing 6A.9.

6.6 Problems

1. Evaluate Fourier series for rectangular pulse represented by Eq. (6.1). Determinecoefficients sn.

2. Evaluate Fourier transform of chirped Gaussian pulse and its spectral half-width �ω

as given by Eq. (6.14).


−50 0 500

0.2

0.4

0.6

0.8

1

1.2

1.4

time [ps]

ampl

itud

e

Fig. 6.16 Evolution of Gaussian pulse by Fourier transform split-step method.

−50

0

50

0

100

2000

0.5

1

1.5

time [ps]distance [km]

Fig. 6.17 Three-dimensional view of the evolution of the Gaussian pulse shown in Fig. 6.16.



157 Appendix 6A



6A.1 rp.m Creates approximations to rectangular pulse6A.2 FT rp.m Evaluates FT of rectangular pulse using analytical expression and

also MATLAB functions6A.3 super gauss.m Plots super Gaussian pulses6A.4 gauss chirp.m Plots chirp Gaussian pulses6A.5.1 ptrain.m Generates sequence of Gaussian pulses6A.5.2 f gauss.m Function used by ptrain.m6A.6 bit gen.m Illustrates NRZ modulation format6A.7 pulse evol.m Evolution of pulses (chirped and unchirped)6A.8.1 pdisp.m Width of Gaussian pulse with dispersion6A.8.2 gauss m.m Function used by pdisp.m. Defines Gaussian pulses6A.9 sslg.m Evolution of Gaussian pulse by Fourier transform split-step method

Listing 6A.1 Function rp.m. Function which draws rectangular pulse and calculatesapproximations based on Fourier series for three values of n.

% File name: rp.m (rectangular pulse)

clear all


% Pulse characteristics

s_0 = 1.0; % pulse amplitude

t_min = 2; % start of pulse

t_max = 10.0; % end of pulse

A = 0.5;

dt = 1/N_max;

t = 0:dt:2*t_max;

x = A*(sign(t-t_min)-sign(t-t_max)); % creates rectangular pulse

hold on

h = plot(t,x);

set(h,’LineWidth’,2);

axis([0 1.2*t_max -0.5 1.7]); % plots rectangular pulse

%

s = 0.0;

for n = [1 3 5]

s = s + 4*s_0/(pi*n)*sin(n*pi*(t-t_min)./(t_max-t_min));

h = plot(t,s);

set(h,’LineWidth’,2);

end

axis([0 1.2*t_max -0.5 1.7]);

xlabel(’time (a.u.)’,’FontSize’,14); % size of x label

ylabel(’s (a.u)’,’FontSize’,14); % size of y label



% Add description

text(5.5, 1.4, ’n = 1’,’Fontsize’,13)

text(7.0, 1.4, ’n = 3’,’Fontsize’,13)

text(8.5, 1.4, ’n = 5’,’Fontsize’,13)

pause

close all

Listing 6A.2 Function FT rp.m. Function plots rectangular pulse, its Fourier transformobtained analytically, and also evaluates its Fourier transform using MATLAB built-infunctions.

% File name: FT_rp.m

% Plots symmetric rectangular pulse of duration tau and its

% Fourier transform

clear all


% Pulse characteristics

tau = 10.0; % 1/2 pulse width

A = 0.5; % pulse height

% Plotting Fourier transform of rectangular pulse using analytical formula

omega = linspace(-20/tau, 20/tau, N_max); % creation of frequency argument

S = 2*A*sin(tau.*omega)./omega;

h = plot(omega,S);

set(h,’LineWidth’,1.5); % thickness of plotting lines

grid

xlabel(’frequency (a.u.)’,’FontSize’,14);

ylabel(’Fourier transform (S) (a.u.)’,’FontSize’,14);


text(0.65, 0, ’A’,’Fontsize’,13);

text(1, 0, ’B’,’Fontsize’,13);

pause

close all

%---- Plotting rectangular pulse -----------------------------------------

dt = 1/N_max;

t = -6*tau:dt:6*tau; % Creation of points for pulse plotting

s = A*(sign(t+tau)-sign(t-tau));

h = plot(t,s);


axis([-2*tau 2*tau 0 2]);

xlabel(’time (a.u.)’,’FontSize’,14); ylabel(’s_r (a.u.)’,’FontSize’,14);


pause

close all

%----- Plotting its Fourier transform using Matlab functions ----------

y = fft(s); % discrete Fourier transform

SM = fftshift(y); % shift zero-frequency component to center of spectrum

N = length(SM);

159 Appendix 6A

k = -(N-1)/2:(N-1)/2;

f = abs(SM);

h = plot(k,f);


axis([-4*tau 4*tau -5000 30000]);

ylabel(’Abs FT of rectangular pulse (a.u)’,’FontSize’,14);

grid


pause

close all

Listing 6A.3 Function super gauss.m. Plots super-Gaussian pulses for several values ofparameter m.

% File name: super_gauss.m

% Plots super-Gaussian pulses

clear all

A = 1;

sigma = 30;

N = 300.0;

t = linspace(-50,50,N);

hold on

for m = [1 2 3 4]

p = A/(sigma*sqrt(2*pi))*exp(-t.^(2*m)/(sigma^(2*m)));

h = plot(p);


end

xlabel(’time’,’FontSize’,14); ylabel(’amplitude’,’FontSize’,14);


pause

close all

Listing 6A.4 Function gauss chirp.m. Generates Gaussian and chirp Gaussian pulses.

% File name: gauss_chirp.m

% Creation of Gaussian pulse and also chirped Gaussian pulse

clear all;

%

t_zero = 20.0; % center of incident pulse

width = 6.0; % width of the incident pulse

C = 15; % chirped parameter

%

N = 300.0;

time = linspace(0,50,N);

pulse = exp(-0.5*((t_zero - time)/width).^2); % Gaussian pulse


pulse_ch = exp(-(0.5+1i*C)*((t_zero - time)/width).^2); % chirped pulse

%

subplot (2,1,1), h = plot(time,pulse);


title(’Gaussian pulse’,’FontSize’,14)



%

subplot (2,1,2), h = plot(time,pulse_ch);


title(’Chirped Gaussian pulse’,’FontSize’,14)



pause

close all

Listing 6A.5.1 Function ptrain.m. Generates a train of pulses.

% File name: ptrain.m

% Generates train of pulses

% Allows for overlap of pulses

clear all

bits = [0 1 0 1 1 0]; % definition of logical pattern

T = 1d-9; % pulse period [s]

num_pulses = length(bits); % number of pulses

N = 1000; % number of time points

width = 0.3*T; % width of pulse

time=linspace(0,num_pulses*T,N);

signal = zeros(1,N);

pulses = zeros(num_pulses,N);

%

t_0= T/2; %position of peak of first impulse in signal

for i=1:num_pulses

if (bits(i)==1)

pulses(i,:)= fgauss(time,width,t_0);

signal = signal + fgauss(time,width,t_0);

end

t_0 = t_0 + T;

end

%

subplot (2,1,1); h = plot(time,pulses);


xlabel(’time’,’FontSize’,14); % size of x label

title(’Individual pulses’,’FontSize’,14)


subplot (2,1,2); h = plot(time,signal);


161 Appendix 6A



title(’Total signal’,’FontSize’,14)

pause

close all

Listing 6A.5.2 Function fgauss.m. Function used by program ptrain.m. Function createsindividual Gaussian pulses.

function pulse = fgauss(time,width,t_0)

% function pulse = wave_gauss(t,T_period)

% Generates individual Gaussian pulse used in the creation of

% train of pulses

pulse = exp(-0.5*((t_0 - time)/width).^2);

Listing 6A.6 Function bits gen.m. Function generates an 8-bits long bit pattern.

% File name: bits_gen.m

%-----------------------------------------------------------------

% Purpose:

% Generates 8-bits long pattern

% generates single bit which is then repeated 8 times

% Source

% R.Sabella and P.Lugli

% "High Speed Optical Communications"

% Kluwer Academic Publishers 1999

% p.32

%*********************************

%

clear all

bits = [0 1 0 0 1 1 0 1]; % Definition of logical pattern

%

T_period = 1d-9; % pulse period [s]

I_bias = 2; % mA

I_m = 3; % mA

tau_r = 0.2*T_period; % rise time

%

% Generation of current pattern corresponding to bit pattern

I_p = 0;

N_div = 50; % number of divisions within each bit interval

t = linspace(0, T_period, N_div); % the same time interval is

% generated for each bit

%******** first bit *************************************

if bits(1)==0, I_p_1 = I_bias + 0*t;

elseif bits(1)==1, I_p_1 = I_bias + I_m + 0*t;

end

%------------------------------------------


% Generates single, arbitrary bit

%t = linspace(0, T_period, N_div);

temp_I = I_p_1;

%

number_of_bits = length(bits);

%

for k = 2:number_of_bits

if bits(k)==1 && bits(k-1)==0

I_p = I_bias + I_m*(1 - exp(-2.2*t./tau_r));

elseif bits(k)==0 && bits(k-1)==1

I_p = I_bias + I_m*exp(-2.2*t./tau_r);

elseif bits(k)==0 && bits(k-1)==0

I_p = I_bias + 0*t;

else bits(k)==1 && bits(k-1)==1

I_p = I_bias + I_m + 0*t;

end

I_p = [temp_I,I_p];

temp_I = I_p;

end

temp_t = t;


t = linspace(0, T_period, N_div);

t = [temp_t,(k-1)*T_period+t];

temp_t = t;

end

%

x_min = 0;

x_max = max(t);

y_min = I_bias;

y_max = I_bias + 1.2*I_m;

%

h = plot(t,I_p);

grid on


xlabel(’time [s]’,’FontSize’,14); % size of x label

ylabel(’Current [mA]’,’FontSize’,14); % size of y label


axis([x_min x_max y_min y_max])

pause

close all

Listing 6A.7 Function pulse evol.m. Function describes evolution width of a chirpedpulse.

% File name: pulse_evol.m

% Describes evolution of pulse width as a function of normalized

163 Appendix 6A

% distance for chirped and unchirped pulses

%

clear all

% Data

beta_2 = -20; % [-20ps^2/km] at 1.55 microns

T_0 = 200; % [ps] = 0.2 ns, bit rate 10Gb/s

L_D = (T_0^2)/abs(beta_2);


x = linspace(0,1.5,N_max); % normalized distance; x = z/L_D

%

hold on

for C = [-1 0 1 2] % chirped coefficient

T = sqrt((1+sign(beta_2)*C*x).^2 + (x).^2);

h = plot(x,T);


end


ylabel(’T_1/T_0’,’FontSize’,14)

xlabel(’z/L_D’,’FontSize’,14)

text(0.35, 1.65, ’C = -1’,’Fontsize’,14)

text(0.35, 1.2, ’C = 0’,’Fontsize’,14)

text(0.35, 0.8, ’C = 1’,’Fontsize’,14)

text(0.35, 0.55, ’C = 2’,’Fontsize’,14)


pause

close all

Listing 6A.8.1 Function pdisp.m. Function describes the evolution of a Gaussian pulsewith dispersion.

% File name: pdisp.m

% Propagation of Gaussian pulse in the presence of dispersion

% using Fourier transform

clear all;

% Creation of input pulse

T_0 = 1.0; % pulse width

T_s = 0.08; % sampling period

t = -4:T_s:4-T_s; % creation of time interval

p = gauss_m(t,T_0);

P=fftshift(fft(p)); % Fourier transform of the original pulse

Fs=1/64; % sampling frequency

N=length(t); % length of time interval

f = -N/2*Fs:Fs:N/2*Fs-Fs; % frequency range

omega = 2*pi*f;

% Parameters of optical fiber


alpha = 0.0; % losses

beta_2 = 60.0; % coefficient beta_2 [ps^2/km]

%beta_2 = 0.0; % coefficient beta_2 [ps^2/km]

beta_3 = 0.01; % coefficient beta_3 [ps^3/km]

% Transfer function of optical fiber

distance = 4000;

H=exp((alpha/2+1i/2*beta_2*omega.^2+1i/6*beta_3*omega.^3)*distance);

%

P_prop = P.*H; % Fourier transform of final pulse

p_prop = ifft(fftshift(P_prop)); % time dependence of final pulse

p_plot = abs(p_prop).^2;

%

h = plot(t, p, ’.’, t, p_plot);



ylabel(’Arbitrary units’,’FontSize’,14); % size of y label


legend(’original pulse’, ’transmited pulse’)

pause

close all

Listing 6A.8.2 Function gauss m.m. Function used by pdisp.m.

function p = gauss_m(t,T_0)

% Definition of Gaussian and super-Gaussian pulses

m = 1; % m=1, usual Gauss; m=3, super-Gauss

%m = 3;

p = exp(-t.^(2*m)/(2*T_0^(2*m)));

Listing 6A.9 Function sslg.m. Function creates a Gaussian pulse in the time domainwhich is subsequently Fourier transformed to the frequency domain and then propagatedover a linear system with dispersion characterized by β2 and β3.

% File name: sslg.m

% Calculates and plots the evolution of a Gaussian pulse in optical fiber

% using Fast Fourier Transform split-step method with linear terms only

clear all

% Input parameters

N= 32; % number of points along time axis

T_domain = 100; % total time domain kept [in ps]

beta_2 = 2; % dispersion coefficient [in ps/nm-km]

beta_3 = 1.01;

%

Delta_t = T_domain/N; % node spacing in time

Delta_om = 2*pi/T_domain; % node spacing in radial frequency

t = Delta_t*(-N/2:1:(N/2)-1); % array of time points

omega = Delta_om*(-N/2:1:(N/2)-1); % array of radial frequency points

165 References

t_FWHM_0 = 20; % initial pulse FWHM [in ps]

P_0 = 1; % initial peak power [in mW]

A_0 = sqrt(P_0); % initial amplitude of the Gaussian

T_0 = t_FWHM_0/(2*sqrt(log(2))); % initial pulse standard deviation

gauss = A_0*exp(-t.^2/(2*T_0^2)); % initial Gauss pulse in time

%

L_D = 200;

z_plot = [0 0.25 0.5 0.75 1.0]*L_D; % z-values to plot [in km]

gauss_F = fftshift(abs(fft(gauss)));% initial Gauss in frequency

z = 0; % starting distance

n = 0; % controls stepping

hold on

for z_val = z_plot % for selected z-values

n = n + 1; % creates new step

% P -propagator function

P = exp(((1j/2)*beta_2*(omega.^2)+(1j/6)*beta_3*(omega.^3))*z_val);

u_F_z = gauss_F.*P; % propagation at point z

u_z = ifft(u_F_z,N); % takes inverse Fourier transform

u_abs_z = abs(u_z).^2;

u = fftshift(u_abs_z); % shifts frequency components

u_3D(:,n) = u’; % create array for 3D plot

plot(t,u,’LineWidth’,1.5)

end

grid on

xlabel(’time [ps]’,’FontSize’,14); ylabel(’amplitude’,’FontSize’,14);


pause

close all

%

% Make 3D plot

for k = 1:1:length(z_plot) % choosing 3D plots every step

y = z_plot(k)*ones(size(t)); % spread out along y-axis

plot3(t,y,u_3D(:,k),’LineWidth’,1.5)

hold on

end

xlabel(’time [ps]’,’FontSize’,14); ylabel(’distance [km]’,’FontSize’,14);


grid on

pause

close all

References

[1] G. P. Agrawal. Fiber-Optic Communication Systems. Second Edition. Wiley, New York,1997.


[2] E. Iannone, F. Matera, A. Mocozzi, and M. Settembre. Nonlinear Optical Communi-cation Networks. Wiley, New York, 1998.

[3] G. P. Agrawal. Nonlinear Fiber Optics. Academic Press, Boston, 1989.[4] G. Keiser. Optical Fiber Communications. Fourth Edition. McGraw-Hill, Boston, 2011.[5] M. M.-K. Liu. Principles and Applications of Optical Communications. Irwin, Chicago,

1996.[6] L. N. Binh. Optical Fiber Communications Systems. CRC Press, Boca Raton, 2010.[7] R. Ramaswami and K. N. Sivarajan. Optical Networks: A Practical Perspective. Second

Edition. Morgan Kaufmann Publishers, San Francisco, 1998.[8] J. C. Cartledge. Improved transmission performance resulting from the reduced chirp of

a semiconductor laser coupled to an external high-Q resonator. J. Lightwave Technol.,8:716–21, 1990.

[9] R. Sabella and P. Lugli. High Speed Optical Communications. Kluwer AcademicPublishers, Dordrecht, 1999.

7 Optical sources

In this chapter, we will give a basic introduction to optical sources with the main emphasis onsemiconductor lasers. The bulk of our description is based on the rate equations approach.We start with a general overview of lasers.

7.1 Overview of lasers

Generic laser structure is shown in Fig. 7.1 [1], [2]. It consists of a resonator (cavity),here formed by two mirrors, and a gain medium where the amplification of electromagneticradiation (light) takes place. A laser is an oscillator analogous to an oscillator in electronics.To form an oscillator, an amplifier (where gain is created) and feedback are needed.Feedback is provided by two mirrors which also confine light. One of the mirrors ispartially transmitting which allows the light to escape from the device. There must bean external energy provided into the gain medium (a process known as pumping). Mostpopular (practical) pumping mechanisms are by optical or electrical means.

Gain medium can be created in several ways. Conceptually, the simplest one is thecollection of gas molecules. Such systems are known as gas lasers. In a gas laser one canregard the active medium effectively as an ensemble of absorption or amplification centers(e.g. like atoms or molecules) with only some electronic energy levels which couple to theresonant optical field. Other electronic states are used to excite or pump the system. Thepumping process excites these molecules into a higher energy level.

A schematic of a pumping cycle of a typical laser is shown in Fig. 7.2. The systemconsists of a ground state level (here labelled as ‘0’), two levels with energies E1 and E2 anda band labelled as ‘3’. The system is pumped by an external source with the frequency ω30.Fast decay takes place between band ‘3’ and energy level ‘2’. In this way the populationinversion is achieved (occupancy of level ‘2’ is larger than level ‘1’), and at some pointlaser action can take place and light with frequency ω21 can be emitted.

The system consisting of level ‘2’ and ‘1’ can be further isolated. Popular visualizationof such systems is known as a two-level system (TLS), see Fig. 7.3. Only two energies (outof many in the case of a molecule) are selected and the transitions are considered withinthose energies. As illustrated, three basic processes are possible: absorption, stimulatedemission and spontaneous emission.

Such TLS are met often in Nature. Generally, for an atomic system, in the case underconsideration, we can always separate just two energy levels, upper level and ground state,thereby forming TLS.

167

168 Optical sources

Resonator (two mirrors)

Gainmedium

Pump

Fig. 7.1 Generic laser structure: two mirrors with a gain medium in between. The twomirrors form a cavity, which confines thelight and provides the optical feedback. One of the mirrors is partially transmitted and thus allows light to escape. Theresulting laser light is directional, with a small spectral bandwidth.

3

0

1

2

30

21Pumptransitions

Fast decay

Fast decay

ωω Laser transitions

Fig. 7.2 Pumping cycle of a typical laser.

Ec

absorption stimulatedemission

spontaneousemission

(a) (c)

hν>Eg Eg~~

(b)

hν hνhν

hν

Ev

Ec

Ev

Ec

Ev

Fig. 7.3 Illustration of possible transitions in two-level system.

As mentioned above, an electron can be excited into the upper level due to externalinteractions (for lasers, process known as pumping). Electrons can lose their energiesradiatively (emitting photons) or non-radiatively, say by collisions with phonons.

For laser action to occur, the pumping process must produce population inversion, mean-ing that there are more molecules in the excited state (here, upper level with energy E2)than in the ground state. If the population inversion is present in the cavity, incoming light

169 Overview of lasers

E2 E2 E 2

E1 E

ρ

1 E1

absorption spontaneousemission

stimulatedemission

(c)(b)(a)

12 1 21( )B N v 21 2 21( )B N v 21 2A Nρ

Fig. 7.4 Illustration of the notation of electron transitions in a TLS.

can be amplified by the system, see Fig. 7.3b where one incoming photon generates twophotons at the output.

The way how TLS is practically utilized results in various types of lasers; like gaseous,solid state, semiconductor. Also, different types of resonators are possible which will bediscussed in subsequent sections.

7.1.1 Transitions in a TLS

Assume the existence of two energy levels E1 and E2 (forming TLS) which are occupiedwith probabilities N1 and N2. Also introduce:

A21 the probability of spontaneous emissionB21 the probability of stimulated emissionB12 the probability of absorption.

The notation associated with those processes is illustrated in Fig. 7.4.We introduce ν21 = (E2 − E1)/h and ρ(ν21) is the density of photons with frequency

ν21. Coefficients A21, B21, B12 are known as Einstein coefficients. The density of photonsρ(ν21) of frequency ν21 can be determined from the Planck’s distribution of the energydensity in the black body radiation as [2]

ρ(ν21) = 8πhν321

c3

1

exp hν21kT − 1

(7.1)

For the laser we require that amplification be greater than absorption. Therefore, thenumber of stimulated transitions must be greater than the absorption transitions. Thus thenet amplification can be created. In the following we will determine the condition forthe net amplification in such systems [2].

The change in time of the occupancy of the upper level is

dN2

dt= −A21N2 − B21N2ρ(ν21) + B12N1ρ(ν21) (7.2)

First term on the right hand side describes spontaneous emission; second term is responsiblefor stimulated emission and the last one for absorption. In thermal equilibrium whichrequires that dN2

dt = 0, using Eq. (7.1) one obtains

N2

{B21

8πhν321

c3(exp hν21

kT − 1) + A21

}= N1B12

8πhν321

c3(exp hν21

kT − 1) (7.3)

170 Optical sources

z = L

rL rR

zz = 0

Fig. 7.5 Schematic illustration of the amplification in a Fabry-Perot (FP) semiconductor laser with homogeneously distributedgain.

Next, assume that N1 and N2 are given by the Maxwell-Boltzmann statistics; i.e.

N1

N2= exp

(−hν21

kT

)(7.4)

Substitute (7.4) into Eq. (7.3) and have

exp

(−hν21

kT

){B21

8πhν321

c3(exp hν21

kT − 1) + A21

}= B12

8πhν321

c3(exp hν21

kT − 1)

From the above equation, the density of photons ρ(ν21) can be expressed in terms ofEinstein coefficients as

8πhν321

c3(exp hν21

kT − 1) = A21

B12 exp( hν21

kT

)− B21

(7.5)

In the above formula, both sides are equal if

B21 = B12 (7.6)

In such case, one finds

A21

B21= 8πhν3

21

c3(7.7)

The relations (7.6) and (7.7) are known as Einstein relations [2].

7.1.2 Laser oscillations and resonant modes

Light propagation with amplification is illustrated in Fig. 7.5. Mathematically it is describedby assuming that there is no phase change on reflection at either end (left and right). Left

171 Overview of lasers

end is defined as z = 0 and right end as z = L. At the right facet, the forward optical wavehas a fraction rR reflected (amplitude reflection) and after reflection that fraction travelsback (from right to left).

In order to form a stable resonance, the amplitude and phase of the wave after a singleround trip must match the amplitude and phase of the starting wave. At arbitrary point zinside the cavity, see Fig. 7.5, the forward wave is

E0egze− jβz (7.8)

where we have dropped eiωt term which is common and defined g = gm − αm, where gm

describes gain (amplification) of the wave and αm its losses. Also rR and rL are, respectively,right and left reflectivities, L length of the cavity and β propagation constant.

The wave travelling one full round will be{E0egze− jβz

} {eg(L−z)e− jβ(L−z)

} {rRegLe− jβL

} {rLegze− jβz

}(7.9)

The above terms are interpreted as follows. In the first bracket there is an original forwardpropagating wave which started at z, in the second bracket there is a wave travelling from zto L, the third bracket describes a wave propagating from z = L to z = 0, and the last onecontains a wave travelling from z = 0 to the starting point z. At that point the wave mustmatch the original wave as given by Eq. (7.8). From the above, one obtains a condition forstable oscillations

rRrLe2gLe−2 jβL = 1 (7.10)

That condition can be split into an amplitude condition

rRrLe2(gm−αm )L = 1 (7.11)

and phase condition

e−2 jβL = 1 (7.12)

From the amplitude condition one obtains

gm = αm + 1

2Lln

1

rRrL(7.13)

From the phase condition, it follows

2βL = 2πn (7.14)

where n is an integer. The last equation determines wavelengths of oscillations since

β = 2π

λn= ωn

c(7.15)

with λn being the wavelength. Typical gain spectrum and location of resonator modes areshown in Fig. 7.6a. Longitudinal modes with angular frequencies ωn−1, ωn and ωn+1 areshown. In time, the mode which has the largest gain will survive; the other modes willdiminish, see Fig. 7.6b.

172 Optical sources

(a)

(b)

gain

n

n

n-1ω ω ω ω

ω ω ω ω

n+1

n+1n-1

Fig. 7.6 Gain spectrum of semiconductor laser and location of longitudinal modes.ωn are FP resonances determined fromphase condition.

(b)

activeregions

Braggmirrors

outputlight

(a)

outputlight

longitudinaldirection

lateraldirection

transversaldirection

Fig. 7.7 VCSEL (left) and in-plane laser (right).

7.2 Semiconductor lasers

A significant percentage of today’s lasers are fabricated using semiconductor technology.Those devices are known as semiconductor lasers. Over the last 15 years or so, severalexcellent books describing different aspects and different types of semiconductor lasershave been published [3], [4], [5], [6], [7].

The operation of semiconductor lasers as sources of electromagnetic radiation is basedon the interaction between EM radiation and the electrons and holes in semiconductors.Typical semiconductor laser structures are shown in Fig. 7.7. Those are: vertical cavitysurface emitting laser (VCSEL) [8] where light propagates perpendicularly to the mainplane and in-plane laser where light propagates in the main plane [9]. The largest dimension

173 Semiconductor lasers

lightout

electrons in

holes in

p-type

n-type

active region

Fig. 7.8 The basic p-n junction laser.

of in-plane structures is typically in the range of 250 µm (longitudinal direction) whereasthe typical diameter of VCSEL cylinder is about 10 µm.

The basic semiconductor laser is just a p-n junction, see Fig. 7.8 where cross-sectionalong lateral-transversal directions is shown. Current flows (holes on p-side and electrons onn-side) along the transversal direction, whereas light travels along the longitudinal directionand leaves device at one or both sides.

In VCSEL, the cavity is formed by the so-called Bragg mirrors and an active regiontypically consists of several quantum well layers separated by barrier layers, see Fig. 7.7.(For a detailed discussion of quantum wells see [1] and [3]. This discussion is beyond thescope of this book.) Bragg mirrors consist of several layers of different semiconductorswhich have different values of refractive index. Due to the Bragg reflection such structureshows a very large reflectivity (around 99.9%). Such large values are needed because a veryshort distance of propagation of light does not allow to build enough amplification whenpropagating between distributed mirrors.

A three-dimensional perspective view of some generic semiconductor lasers is shownin Fig. 7.7. The structures consists of many layers of various materials, each engaged in adifferent role. Those layers are responsible for the efficient transport of electrons and holesfrom electrodes into an active region and for confinement of carriers and photons so theycan strongly interact. Modern structures contain so-called quantum wells which form anactive region and where conduction-valance band transitions are taking place. It is possibleto have different types of mirrors as they also provide mode selectivity. Two basic types areillustrated in Fig. 7.9 and Fig. 7.10 [9]. They are known as distributed feedback (DFB) anddistributed Bragg (DBR) structures.

In DFB lasers, grating (corrugation) is produced in one of the cladding layers, thuscreating Bragg reflections at such periodic structure. The structure causes a wavelengthsensitive feedback. It should be emphasized that the grating extends over the entire laserstructure. When one restricts corrugation to the mirror regions only and leaves a flat activeregion in the middle, then the so-called DBR structure is created, see Fig. 7.10, which alsoprovides wavelength sensitive feedback.

174 Optical sources

GratingActivelayer andwaveguide

Fig. 7.9 Basic DFB laser structure.

DBR mirror DBR mirror

Activelayer andwaveguide

Fig. 7.10 Basic DBR laser structure.

Conductionband

Valenceband

Ec

Ecv Scv,sp Scv,st Svc,abs

EFc

EFv

Ev

Fig. 7.11 Electron transitions between conduction and valence bands.

7.2.1 Electron transitions in semiconductors

Previous discussion of TLS will now be extended to describe transitions between bands insemiconductors, see Fig. 7.11. For that purpose, we introduce:

• fv the probability of the state of energy Ev in the valence band being filled.• fc the probability of the state of energy Ec in the conduction band being filled.

Rates of transitions can now be determined, similarly to the TLS description. Here weuse index notation appropriate to semiconductors, i.e. c, v instead of 1, 2. The rates are

Sspon = Scv,sp = A21 fc (1 − fv ) (7.16)

Sstim = Scv,st = B21 fc (1 − fv ) ρ(Ecv ) (7.17)

Sabs = Scv,abs = B12 fv (1 − fc) ρ(Ecv ) (7.18)


+

++++ +

++ +

n-typep-type

p-type

n-type

(a)

(b)

EFv

EFv

Ev

Ev

EFc

EFc

Ec

Ec

Fig. 7.12 Energy-band diagram of a p-n junction. (a) thermal equilibrium, (b) forward bias.

Fermi-Dirac statistics are assumed for the appropriate probabilities as

fv = 1

exp(

Ev−EFv

kT + 1) (7.19)

fc = 1

exp(Ec−EFc

kT + 1) (7.20)

Here EFc and EFv are the quasi-Fermi levels for electrons and holes.The condition for stimulated emission to exceed absorption is

Sstim > Sabs

which gives

B21 fc (1 − fv ) > B12 fv (1 − fc)

Since the coefficients B21 and B12 are equal, after substituting expressions (7.19) and (7.20)one obtains

exp

(Ev − EFv

kT

)> exp

(Ec − EFc

kT

)which can be written in the form

EFc − EFv > Ec − Ev = hν (7.21)

The above inequality is known as Bernard-Duraffourg condition [10].

7.2.2 Homogeneous p-n junction

Energy-band diagrams of a p-n junction in thermal equilibrium and under forward biasconditions are shown in Fig. 7.12.

176 Optical sources

Electrons at the n-type side do not have enough energy to climb over the potential barrierto the left. Similar situation exists for holes in the p-type region, see Fig. 7.12a. One needsto apply forward voltage to lower the potential barriers for both electrons and holes, seeFig. 7.12b.

An application of the forward voltage also separates Fermi levels. Thus two differentso-called quasi-Fermi levels EFc and EFv are created which are connected with an externalbias voltage as

EFc − EFv = eVbias (7.22)

With the lowered potential barriers, electrons and holes can penetrate central region wherethey can recombine and produce photons. However, the confinement of both electrons andholes into the central region is very poor (there is no mechanism to confine those carriers).Also, there is no confinement of photons (light) into the region where electrons and holesrecombine (the region is known as an active region). Therefore, the interaction betweencarriers and photons is weak, which makes homojunctions a very poor light source. Onemust, therefore, provide some mechanism which will confine both carriers and photons intothe same physical region where they will strongly interact. Such a concept is possible withthe invention of heterostructures, which will be discussed next.

7.2.3 Heterostructures

A heterojunction is formed by joining dissimilar semiconductors. The basic type is formedby two heterojunctions and it is known as a double-heterojunction (more popular name isdouble-heterostructure).

The materials forming double heterostructures have different bandgap energies anddifferent refractive indices. Therefore, in a natural way potential wells for both electronsand holes are created. Schematic energy-band diagrams of a double-heterostructure p-njunction in thermal equilibrium and under forward bias conditions are shown in Fig. 7.13.The bandgap and refractive index for InGaAsP material will now be summarized.

Bandgap

For the In1−xGaxAsyP1−y system, the relation between compositions x and y which resultsin the lattice-match to InP is

y = 0.1894y

0.4184 − 0.013y(7.23)

In such case the bandgap is [11], [12]

Egap[eV ] = 1.35 − 0.72y + 0.12y2 (7.24)

with the extra relations y ≈ 2.20x and 0 ≤ x ≤ 0.47. The material for such compositionsthus covers bandgaps in the range of wavelengths 0.92 µm ⇔ 1.65 µm. For example, thematerial In0.74Ga0.26As0.57P0.43 (i.e. x = 0.26, y = 0.57) has bandgap Eg = 0.97eV, whichcorresponds to the wavelength λ = 1.27 µm.


Table 7.1 Typical values of refractive indices for various compositions.

Wavelength Range of y compositions Refractive index

λ = 1.3 µm 0 ≤ y ≤ 0.6 n(y) = 3.205 + 0.34y + 0.21y2

λ = 1.55 µm 0 ≤ y ≤ 0.9 n(y) = 3.166 + 0.26y + 0.09y2

+

+ ++++

++ +

n-typep-type

p-type Active

Active

n-type

(a)

(b)

EFc

EFc

Ev

Ev

EFv

EFv

Ec

Ec

Fig. 7.13 Energy-band diagram of a double-heterostructure p-n junction. (a) Thermal equilibrium, (b) forward bias.

Refractive index

As was explained before, the dependence of refractive index as a function of wavelengthis described by Sellmeier equation. For important telecommunication wavelengths, namely1.3 µm and 1.55 µm, the y dependence of refractive index of In1−xGaxAsyP1−y that is latticematched to InP is shown in Table 7.1 [13].

In the wells formed by heterostructures, electrons and holes recombine thus generatinglight. Also, differences in the values of refractive indices such that central region haslarger value of refractive index create planar optical waveguide where light propagates andefficiently interacts with carriers.

Some illustrative values of bandgaps and refractive indices for InGaAsP/InP heterostruc-ture are shown in Fig. 7.14.

7.2.4 Optical gain

For efficient and reliable numerical simulations, an exact mathematical expression of thematerial gain is critical. Such an expression should also be subject to an experimentalverification. This problem has been investigated since the early developments of semicon-ductor lasers [14], [15], [16]. The first step is usually the determination of gain spectra

178 Optical sources

1.35eV 0.95eV

Refractiveindex

energy

activelayer

p-InPn-InP InGaAsP

3.53.2

Fig. 7.14 Band structure and refractive index for InGaAsP/InP heterostructure.

and its comparison with experiment. The simplest approach will be provided in the nextsection. Typical curves of gain spectra for a four quantum well system and comparisonwith experimental measurements are shown in Fig. 7.15, from [17]. Mathematics involvedin determining optical gain of semiconductor quantum-well structures is complex and hasbeen discussed extensively, for example [18], [19], [20]. As a result, analytic approxima-tions of the optical gain which can be used in fast calculations were determined, see [21],[22], [23]. From an extensive discussion it was determined that the peak material gain forbulk materials varies linearly with carrier density

g(N, λ) = a(λ) min (N, P) − b(λ) (7.25)

The parameter a(λ) is commonly called the differential gain.On the basis of experimental observations, Westbrook [24], [25] extended the linear gain

peak model to allow for wavelength dependence. In its simplest form it can be written as[24], [26], [27]

g(N, λ) = a(λp)N − b(λp) − ba

(λ − λp

)2(7.26)

where λp is the wavelength of the peak gain and ba governs the base width of the gainspectrum. Wavelength peak λp can be also carrier density dependent.

Gain (absorption) in a semiconductor is a function of carrier density n [cm−3] (seeFig. 7.16, which shows gain spectrum for several values of carrier concentration) [28].When n is below transparency density ntr, the medium absorbs optical signal. For n > ntr,

optical gain in the material exceeds loss. The dependence of the so-called gain peak on thecarrier density for quantum well systems is logarithmic, see Fig. 7.17. It is described by theexpression [4]

g = g0 lnN

Ntr(7.27)

For modelling purposes, a linear approximation is often employed. In the above formulasg0 is the differential gain, Ntr transparency carrier density and Nth is the carrier density atthreshold.


0.830.820.810.800.790.78-40

-30

-20

-10

0

10

20

30 Theoretical Experimental

25 mA

21 mA

Mod

al G

ain

(/cm

)

Photon Energy (eV)

Fig. 7.15 Fitted theoretical and experimental net modal gain versus photon energy including leakage terms for three values ofinjected current, from [17].

Gain(×104m-1)

3×1018 cm-1

2×1018 cm-1

1×1018 cm-1

1450-2

0

2

4

6

8

1500

Wavelength [nm]

1550 1600

Fig. 7.16 Schematic optical gain spectra for various carrier densities.

180 Optical sources

NNthNtr

g

thg

g

thg

Ntr thN N

Fig. 7.17 Illustration of threshold and transparency densities for linear (left) and logarithmic (right) gain models.

7.2.5 Determination of optical gain

The simplest approach to determine optical gain is based on Fermi golden rule. Here wederive it for T = 0. Transition rate between two levels a and b is [1]

Wab = 2π

�

∣∣H′ab

∣∣2 δ (Eb − Ea − �ω) (7.28)

where H′ab is the matrix element describing transitions between levels a and b. Assume

parabolic bands for both electrons and holes with the effective mass m∗ taking the value ofm∗

c for conduction band and m∗v for a valence band as follows:

E (k) = �2k2

2m∗ (7.29)

From the above

Eb − Ea = �2k2

2

(1

mc+ 1

mv

)+ Eg (7.30)

Due to conservation of the crystal momentum in the transition, one has �kin = �k f i [29]. Fora single transition at specific value of k, one therefore has

W (k) = 2π

�

∣∣H ′vc (k)2

∣∣ δ (�2k2

2mr+ Eg − �ω

)(7.31)

where mr = mvmc

mv+mcis reduced effective mass. If N is a total number of transitions per second

in crystal volume V and g (k) = k2Vπ2 is number of states per unit k in volume V , the total

number of transitions per second is

N =∞∫

0

W (k) · g (k) dk

= 2V

π�

∞∫0

∣∣H ′vc (k)

∣∣2 δ

(�

2k2

2mr+ Eg − �ω

)k2dk (7.32)

181 Rate equations

To evaluate integral, let us introduce new variable X ≡ �2k2

2mr+ Eg − �ω. Integral takes the

form

N = 2v

π�

∫ ∣∣H ′vc (k)

∣∣2 mr

�δ (X )

√2mv

�2

(X + �ω − Eg

)dX

= V

π

∣∣H ′vc (k)

∣∣2 (2mr)3/2

�4

(�ω − Eg

)1/2(7.33)

where �2k2

2mr+ Eg = �ω. We are in a position now to determine absorption coefficient α. It

is defined as

α (ω) ≡ power absorbed per unit volume

power crossing a unit area= (N ) · (�ω) /V

ε0nE20 c/2

(7.34)

where n is refractive index, c is the velocity of light in a vacuum and E0 is the amplitude ofelectric field. Use

H ′vc (k) = eE0χvc

2, χvc = 〈uvk′ |x| uck〉 (7.35)

Using the above and expression for N , one finds

α0 (ω) = ωe2χ2vc (2mr)

3/2

2πε0nc�3

(�ω − Eg

)1/2 = K(�ω − Eg

)1/2(7.36)

Example Estimates of K.Data for GaAs. �ω = 1.5 eV, mv = 0.46m0, mc = 0.067m0, χvc = 3.2 A,n = 3.64.

Those data give K ≈ 11 700 cm−1 (eV )−1/2 , and α0 (ω) = 1170 cm−1 [1].At T = 0 depending on the value of �ω, one has the following possibilities summarized

next:

�ω < Eg α (ω) = 0Eg < �ω < EFc − EFv α (ω) = −α0 (ω) = −K(�ω − Eg)

1/2 amplificationEFc − EFv < �ω α (ω) = α0 (ω) = K(�ω − Eg)

1/2 absorption(7.37)

7.3 Rate equations

With all basic elements in place we are now ready to provide the simplest (phenomenolog-ical) description of semiconductor lasers. It is based on rate equations [4]. The main rolein those devices is played by two subsystems: carriers (electrons and holes) and photons.They interact in the so-called active region, defined as part of the structure where recom-bining carriers contribute to useful gain and photon emission. We describe both subsystemsseparately, starting with carriers.

In Fig. 7.18 (adapted from [4]), we schematically showed model of a laser operatingbelow threshold. It resembles a tank partially filled with water continuously flowing in andat the same time water leaves the tank. The water models carriers which are continuously

182 Optical sources

current

RnrRsp

Rloss

leakagecurrent

Fig. 7.18 Model of a laser operating below threshold. From L. A. Coldren and S. W. Corzine, Diode Lasers and Photonic IntegratedCircuits, Wiley (1995). Reprinted with permission of John Wiley & Sons, Inc.

current

RnrRsp

RstNthleakagecurrent Rloss

Fig. 7.19 Model of a laser operating above threshold. From L. A. Coldren and S. W. Corzine, Diode Lasers and Photonic IntegratedCircuits, Wiley (1995). Reprinted with permission of John Wiley & Sons, Inc.

provided by current flowing in. Not all the current at electrodes actually reaches the device(tank). Some of it is lost as so-called leakage current. Below threshold, carriers disappearthrough losses in the device (Rloss), non-radiative recombination (Rnr) and spontaneousemission (Rsp). Above threshold (shown schematically in Fig. 7.19), the tank is completelyfilled with water (in laser the situation corresponds to the so-called threshold with densityNth). Its operation is mostly dominated by yet another process, namely stimulated emission(here characterized by coefficient Rst).

Next, based on the above picture we will formulate equations describing dynamics ofcarriers and photons.

7.3.1 Carriers

Inside the laser there exist two types of carriers: electrons and holes. Both are described ina similar way; however, parameters used will be different for both types.

The rate of change of carrier density is governed by

dN

dt= Ggener − Rrecom (7.38)

where the terms on right are responsible for generation and recombination of carriers andare given by

Ggener = ηiI

q · V

with V being the volume of the active region, q is the electron’s charge and I is the electriccurrent. The internal efficiency ηi describes the fraction of terminal current that generates

183 Rate equations

carriers in the active region. The recombination term consists of several contributions

Rrecom = Rsp + Rnr + Rl + Rst

The meaning of terms is as follows:

Rsp − spontaneous recombination rateRnr − nonradiative recombination rateRl − carrier leakage rateRst − net stimulated recombination rate

Recombination processes are described phenomenologically as [4]

Rrecom = N

τ+ vgg(N )S (7.39)

7.3.2 Photons

Let S be the photon density. We postulate the following rate of change of photon density:

dS

dt= �Rst − S

τp+ �βspRsp

with the following definitions:

τp – photon life-timeβsp – spontaneous emission factor (reciprocal of the number of optical modes)Rst – stimulated recombination.� – confinement factor which is the ratio of the active layer volume to the volume of the

optical mode.

Consider growth of a photon’s density over the active region, (assume � = 1)

S + �S = Seg·�z

where g is gain.

if �z � 1 then exp (g · �z) ≈ 1 + g�z

Using the relation �z = vg�t (vg−group velocity), one finds S = Sg · vg · �t.Thus, the generation term can be written as(

dS

dt

)gen

= Rst = �S

�t= vggS

Finally, the rate equations used in this section are

dN

dt= ηi

I

qV− N

τ− vgg(N )S (7.40)

dS

dt= �vgg(N )S − S

τp+ �βspRsp (7.41)

We have explicitely indicated that gain g depends on a carrier’s concentration.

184 Optical sources

7.3.3 Rate equation parameters

Other important parameters which in the simple model can be taken as constants, in facthave complicated dependencies. Those parameters are:

1. the carrier’s lifetime τ which strongly depends on carrier’s density. Typical dependenceis shown below:

1

τ= A + BN + CN2

where coefficient A describes non-radiative processes, B is responsible for spontaneousrecombination and C describes non-radiative Auger recombinations.

2. photon lifetime τp which is

1

vgτp= αi + αm = �gth

Here αm describes mirror reflectivity

αm = 1

Lln

1

R

and αi account for all losses.The meaning of all parameters appearing in rate equations and gain models is summarized

in Table 7.2 along with typical values for those symbols. The parameters were collectedfrom [30], [31] and [32].

In the following, rate equations will provide a starting point for analysis of dynamicalproperties of semiconductor lasers. Before we start detailed analysis based on rate equations,we will establish a rate equation for electric field.

7.3.4 Derivation of rate equation for electric field

The relevant equation will be derived starting from the wave equation. We concentrate onFabry-Perot lasers. The starting wave equation is

∇2E(r, t) − 1

c2

∂2

∂t2E(r, t) = μ0

∂2

∂t2P(r, t) (7.42)

Assume the following decomposition:

E(r, t) = aEt (x, y) sin (βzz) E(t)e jωt (7.43)

where a is a unit vector describing polarization, βz propagation constant in the z-direction.The transversal field Et (x, y) obeys the following equation:(

∂2

∂x2+ ∂2

∂y2

)Et (x, y) = −κ2

t Et (x, y) (7.44)

It is further assumed that E(t) is slowly varying compared to e jωt . Substituting solution(7.43) into (7.42), neglecting fast-varying terms in time and using (7.44), one obtains{

κ2t − β2

z − 1

c2

[2 jω

∂E(t)

∂t− ω2E(t)

]}aEt (x, y) sin (βzz) e jωt = μ0

∂2

∂t2P(r, t)

185 Rate equations

Table 7.2 Basic parameters appearing in rate equation approach and their typical values.

Symbol Description Value and unit

N carrier density cm−3

S photon density cm−3

I current mAq elementary charge 1.602 × 10−19CL cavity length 250 µmw width of active region 2 µmd thickness of active region 80Aηi fraction of the injected current

I into active region 0.8Vactive volume of active region L · w · dτ carrier lifetime 2.71nsvg group velocity c/nref

nre f refractive index 3.4τp photon lifetime 2.77 ps� confinement factor 0.01βsp spontaneous emission factor 10−4

a differential gain (linear model) 5.34 × 10−16 cm2

Ntr carrier density at transparency 3.77 × 1018 cm−3

Ith current at threshold 1.11 mAαm facet loss 45 cm−1

λph laser wavelength 1.3 µm

To evaluate terms on the right hand side, we need to account for a nonlocal relation betweenpolarization and susceptibility:

P(r, t) = ε0

∫χ(r, t ′)E(r, t − t ′)dt ′ (7.45)

In the above E(r, t −t ′) is given by (7.43). As E(t) is slowly varying in time, we can expandit into Taylor series around t. The general formula for Taylor expansion of the functionf (x) around a is f (x) = f (a) + df

dx (x − a). In our case, substituting x = t − t ′ and a = t,we obtain

E(t − t ′) ≈ E(t) + dE(t)

dt(−t ′)

Full electric field is therefore

E(r, t − t ′) = Et (x, y) sin (βzz) E(t − t ′)e jω(t−t ′ )

Substituting the last result and Taylor expansion for E(t−t ′) into expression for polarization(7.45), one finally obtains the wave equation for E(t):[

−β20 + ω2

c2εr(ω)

]E(t) − 2 jω

c2

[εr(ω) + 1

2ω

dεr(ω)

dω

]dE(t)

dt= 0 (7.46)

186 Optical sources

Here εr(r, ω) = 1 + χ(r, ω) and Fourier transform of susceptibility is (we have dropped rdependence)

χ(ω) =∫

dt ′χ(t ′)e− jωt ′

Susceptibility is further separated into terms which account for various physical effects asfollows:

χ(ω) = 1 + χb + χp − jχloss

where χb = χ ′b + jχ ′′

b is due to background, χp = χ ′p + jχ ′′

p is induced by pump andχloss accounts for losses. The relative dielectric constant is written as (dropping argumentdependencies) εr = ε′

r + jε′′r . The real part of εr is approximated in terms of refractive

index ε′r = (n0 + �np)

2 ≈ n20 + 2n0�np, where n0 is the background refractive index and

�np is the change induced by pump. Using the above results, εr takes the form

εr = ε′r + jε′′

r ≈ n20 + 2n0�np + jε′′

r

= 1 + χ ′b + χ ′

p + j(χ ′′

b + χ ′′p − χloss

)(7.47)

Explicitly, one has the following identifications:

1 + χ ′b = n2

0

χ ′p = 2n0�np

χ ′′b + χ ′′

p − χloss = ε′′r

Using the above relations, the term in the equation for slowly varying amplitude E(t),Eq. (7.46) is

ω2

c2εr(ω) − β2

0 = ω2 − ω20

c2n2

0 + ω2

c2χ ′

p + jω2

c2

(χ ′′

b + χ ′′p − χloss

)(7.48)

where we have used the approximate relation

β0 ≈ ω0

cn0

Imaginary parts of background and pump susceptibilities are related to gain via an experi-mental relation

ω

cn0

(χ ′′

b + χ ′′p

)= �g(N )

where g(N ) is gain. Assume also

χloss = ω

cn0αloss

With the last two relations, term given by Eq. (7.48) can be expressed as

ω2

c2εr(ω) − β2

0 = ω2 − ω20

c2n2

0 + ω2

c22n0�np + j

ωn0

c[�g(N ) − αloss]

187 Analysis based on rate equations

Using n2(ω) = εr(ω), the dispersion term in (7.46) is evaluated as

εr(ω) + 1

2ω

dεr(ω)

dω= n2(ω) + 1

2ωn(ω)

dn(ω)

dω= n(ω) + ωng(ω) (7.49)

where we have defined group index ng(ω) as

ng(ω) = n(ω) + ωdn(ω)

dω(7.50)

Using the results (7.48) and (7.49), Eq. (7.46) is

dE(t)

dt={− j (ω − ω0)

n0

ng− j

ω

ng�np + 1

2vg [�g(N ) − αloss]

}E(t) (7.51)

where we approximated ω2 − ω20 = (ω − ω0) (ω + ω0) � 2ω (ω − ω0) and also n0(ω)

n(ω)� 1

and used the relation cng

= vg. Our Eq. (7.51) is very similar to Eq. (6.2.9) of Agrawal andDutta [9].

7.4 Analysis based on rate equations

Rate equations just introduced will now be used to study some dynamical properties ofsemiconductor lasers. We start with steady-state.

7.4.1 Steady-state analysis

In this situation one has no change in time; i.e.

dN

dt= dS

dt= 0

Rate equations take the form

ηiI0

qV− N0

τ− vgg(N0)S0 = 0 (7.52)

and

�vgg(N0)S0 − S0

τp+ �βspRsp = 0 (7.53)

If in the last equation we neglect the small spontaneous emission term, we obtain anexpression for a photon life-time

1

τp= �vgg(N0) (7.54)

188 Optical sources

7.4.2 Small-signal analysis with the linear gain model

When neglecting spontaneous emission and using linear gain model where gain is given byg = a (N − Ntr), one obtains

dN

dt= ηi

I

qV− N

τ− vga (N − Ntr) S (7.55)

anddS

dt= �vga (N − Ntr) S − S

τp(7.56)

Assume that all time-dependent quantities oscillate as

I = I0 + i(ω)e jωt

N = N0 + n(ω)e jωt

S = S0 + s(ω)e jωt

where ω is the angular frequency of an external (small) perturbation and I0 is the bias valueof current. Here N0 and S0 are the solutions in the steady-state. Substitute into the first rateequation and after multiplication and neglecting second-order term (n(ω)s(ω)) one obtains

jωn(ω)ejωt = ηi

qV

[I0 + i(ω)e jωt

]− N0

τ− 1

τn(ω)e jωt

−vga((N0 − Ntr) S0 + (N0 − Ntr) s(ω)e jωt+S0n(ω)e jωt

)+ O(n2)

Using steady-state results, expression (7.54) for photon life-time and dropping e jωt depen-dence, one finally obtains

jωn(ω) = ηi

qVi(ω) − n(ω)

τ− s(ω)

�τp− vgaS0n(ω) (7.57)

The second equation for photons is obtained in a similar way. Substituting small-signalexpressions, neglecting the second-order term, using the photon life-time expression andfinally dropping e jωt dependence, one obtains

jωs(ω) = �vgaS0n(ω) (7.58)

From the above equations, we want to determine modulation response which is defined as[33], [34]

M (ω) = s(ω)

i(ω)

Expressing n(ω) from Eq. (7.58) and substituting into (7.57), one obtains

s(ω)

i(ω)=

ηi

eV �vgaS0

D(ω)

where D(ω) is given by

D(ω) = −ω2 + jω

(1

τ+ vgaS0

)+ vgaS0

τp


Define response function as

r(ω) =∣∣∣∣ s(ω)

i(ω)

∣∣∣∣ =ηi

eV �vgaS0

|D(ω)|If we write D(ω) = a + jb, we have |D(ω)|2 = a2 + b2. Therefore

|D(ω)|2 =(

ω2 − vgaS0

τp

)2

+ ω2

(1

τ+ vgaS0

)2

We will see soon that the response function has a peak at frequency fR. To find thatfrequency, we evaluate first derivative ∂|D(ω)|2

∂ω2 and set it to zero. Explicitly

∂|D(ω)|2∂ω2

∣∣∣∣ω=ωR

= −2

(vgaS0

τp− ω2

R

)− 1

2

(1

τ+ vgaS0

)= 0

From the above

ω2R = vgaS0

τp− 1

2

(1

τ+ vgaS0

)(7.59)

which is known as a relaxation-oscillation frequency and it describes the rate at which energyis exchanged between photon system and carriers. The second term in (7.59) is usually verysmall and can be neglected. Relaxation-oscillation frequency is thus approximated as

ωR =√

1

τp

S0vga

1 + εS0(7.60)

To estimate the typical values of the relaxation-oscillation frequency, let us assume [1]:L = 300 µm, τph ≈ 10−12s and τ ∼ 4 × 10−9s. One finds AP0 ∼ 109 s−1. So, thezeroth-order expression is a very good one.

Using MATLAB code and parameters shown in Listings 7A.1.1 and 7A.1.2, one obtainsthe results shown in Fig. 7.20.

7.4.3 Small-signal analysis with gain saturation

Let us remind ourselves of the rate equations which were established in the previous section:

dN

dt= ηi

I

qV− N

τ− vgg(N )S (7.61)

dS

dt= �vgg(N )S − S

τp+ βspRsp (7.62)

Use small-signal assumptions

I(t) = I0 + i(t)

N (t) = N0 + n(t)

S(t) = S0 + s(t)

Also, generalize the gain model to include saturation as follows:

g(n) = g0 + g′ (N − N0)

1 + εS

190 Optical sources

10−2

10−1

100

101

102

−20

−10

0

10

20

30

frequency [GHz]

Res

pons

e fu

ncti

on [

dB]

10 μW

1 mW

10 mW

0.1W

1W

Fig. 7.20 Modulation characteristics of semiconductor lasers from small-signal analysis.

where g′ = ∂g∂N

∣∣∣N=N0

is differential gain evaluated at N = N0 and g0 = g(N0). The factor

1 + εS introduces nonlinear gain saturation. The gain compression parameter ε has a smallvalue and the term εS is small compared with the one even at very high optical power. Theeffect of this term on dc properties is small and can be neglected. However, it significantlyaffects dynamics of a semiconductor laser. The reason is that the laser’s dynamics dependson the small difference between gain and cavity loss. This difference is only about a fewpercent. Therefore, even small gain compression due to ε produces significant effect.

Evaluate the gain compression term as follows:

1 + εS = 1 + ε (S0 + s(t)) = (1 + εS0)

(1 + εs(t)

1 + εS0

)With the above result, gain can be approximated as

g(n) = g0

1 + εS0+ g′n(t)

1 + εS0− g0

1 + εS0

εs(t)

1 + εS0(7.63)

Using the above results, stimulated emission term takes the form

g(n)S = g0S0

1 + εS0+ g′S0

1 + εS0n(t) − g0S0

(1 + εS0)2εs(t) + g0

1 + εS0s(t) + O(s2) (7.64)

As usual, the following analysis is separated into two parts: dc and ac analysis.

dc analysisWith the assumption that d

dt = 0 from rate equations, one obtains

ηiI0

qV− N0

τ− vg

g0S0

1 + εS0= 0


and

�vgg0S0

1 + εS0− S0

τp+ βspRsp = 0

Neglecting spontaneous emission (βsp = 0), from the second equation one obtains anexpression for the photon life-time τp:

1

τp= �vg

g0

1 + εS0(7.65)

ac analysisSubstitute small-signal assumptions into rate equations, use an approximation for stim-

ulated emission (7.64) and eliminate dc terms. The results are

dn(t)

dt= ηi

i(t)

qV− n(t)

τ− vgg′S0

1 + εS0n(t) + 1

τp�

S0

1 + εS0εs(t) − 1

τp�s(t)

ds(t)

dt= �vgg′S0

1 + εS0n(t) − 1

τp

S0

1 + εS0εs(t)

In the above equations we used expression (7.65). Those equations can be written in matrixform:

d

dt

[n(t)s(t)

]+[

A B−C D

] [n(t)s(t)

]=[ηi

i(t)qV

0

]where

A = 1

τ+ vgS0g′

1 + εS0, B = 1

τp�− 1

τp�

S0

1 + εS0ε, C = �vgS0g′

1 + εS0, D = 1

τp

S0

1 + εS0ε

Assuming harmonic time dependence of the form exp ( jωt), the matrix equation is[jω + A B−C jω + D

] [n(t)s(t)

]=[ηi

i(t)qV

0

]Solving the above system, one obtains modulation response which is expressed as

s(t)

i(t)= ηi

1

qV

C

H (ω)

where

H (ω) = ( jω + A) ( jω + D) + CB

The function H (ω) can be written as

H (ω) = −ω2 + jωγ + ω2R

Here ωR is the relaxation-oscillation frequency and γ is the damping factor. One definesmodulation response r(ω) as [30]

r(ω) = H (ω)

H (0)

192 Optical sources

10−1

100

101

102

−20

−15

−10

−5

0

5

10

15

20

frequency [GHz]

Res

pons

e fu

ncti

on [

dB]

= 0

= 1.5 × 10−17 = 5 × 10−17

Fig. 7.21 The effect of gain compression on modulation response.

The plot of response r(ω) is shown in Fig. 7.21 for three values of ε. The effect of gaincompression parameter ε is clearly visible. MATLAB code used to obtain those results isprovided in Listings 7A.2.1 and 7A.2.2.

7.4.4 Large-signal analysis for QW lasers

Large-signal analysis based on rate equations for semiconductor lasers has been reportedextensively [35], [36], [37]. Main equations without spontaneous emission are

dN

dt= ηi

I

eV− N

τ− vgg (N ) · S (7.66)

dS

dt= � · vgg (N ) · S − S

τp(7.67)

All symbols have been explained previously and their values are summarized in Table 7.2.Typical results for step current are shown in Fig. 7.22. MATLAB code used to obtain

those results is provided in Appendix, Listings 7A.3.1 and 7A.3.2.One can observe that when we neglect spontaneous emission term in the rate equation

for photons, photon density tends to zero (steady-state value) as t −→ ∞, the result whichis also evident from the steady-state analysis.

7.4.5 Frequency chirping

Frequency chirping arises during direct modulation of semiconductor laser when carrierdensity undergoes abrupt change. As a result, material gain also changes. This changeresults in a variation of the refractive index, which in turn affects the phase of the electricfield.


0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

time [ns]

Arb

itra

ry u

nits

carrier densityphoton density

Fig. 7.22 Numerical solutions of large-signal rate equations after applying rectangular current pulse at t = 0. Normalizedvalues of electron density N(t) and photon density S(t) are shown.

The rate of change of the phase is

dφ

dt= 1

2vαH (�g − αloss)

An associated change in frequency is created which is described by

�ν(t) = 1

2π

dφ

dt= 1

4πvαenh (�g − αloss)

This change in frequency is called ‘chirp’. During direct modulation it results in a shift infrequency of the order of 10–20 GHz.

To eliminate chirp effect, an external modulator should be used. In such cases a laserproduces constant output power and a separate modulator provides modulation.

Another way to reduce chirp is to fabricate devices with small values of line-widthenhancement factors αH .

αH ≡ −dnrdNdni

dν

where n = nr + jni (complex refractive index).

7.4.6 Equivalent circuit models

Often in engineering practice there is a need to operate with circuit models, instead of morecomplicated physical-based models. Here we outline the method of constructing such amodel for a semiconductor laser.

194 Optical sources

7.4.7 Equivalent circuit model for a bulk laser

Equivalent circuit models of semiconductor lasers provide further understanding of thelaser properties. They are derived from rate equations. Development of those models wasinitialized a long time ago [38], [39], [40]. More recent works which also include additionaleffects like carrier transport or proper representation of optical gain, are [41], [42], [43]. Inthe following, we outline the creation of equivalent circuit model for a bulk semiconductorlaser following Kibar et al. [41].

For bulk lasers, rate equations are [41], [9]

dS

dt= (G − γ ) S + Rsp (7.68)

dN

dt= I

q− γeN − G · S (7.69)

Here N, S are the total number of electrons inside the cavity and total number of photons inthe lasing mode. Those are dimensionless quantities. We introduce small deviations fromequilibrium as

S(t) = S0 + δS(t) (7.70)

N (t) = N0 + δN (t) (7.71)

Optical gain is approximated as

G = G0 + ∂G

∂NδN + ∂G

∂SδS = G0 + GNδN + GSδS (7.72)

In the steady-state one obtains

0 = (G0 − γ ) S0 + Rsp,0

0 = I0

q− γeN0 − G0 · S0

Observing that spontaneous emission Rsp = Rsp(N ) depends on N , and performing anexpansion

Rsp = Rsp,0 + ∂Rsp

∂NδN (7.73)

one can derive small-signal equations

dδS

dt= −�SδS + σNδN (7.74)

dδN

dt= −�NδN − σSδS (7.75)

where

�S = Rsp

S0− S0GS, �N = γe + N0

∂γe

∂N+ S0GN (7.76)

and

σN = S0GN + ∂Rsp

∂N, σS = G0 + S0GS (7.77)


C

L

RN

RS

iLiCiR

Fig. 7.23 Equivalent circuit model for a bulk laser.

Using these equations, one creates equivalent circuit model. It is based on an observationthat the small-signal rate equations are similar to the voltage and current equations ofthe RLC circuit. We therefore assume (following Kibar et al., [41]) that electrons in alaser cavity can be represented by the charge across a capacitor and the photons arerepresented by the magnetic flux of an inductor. Formally one introduces the followingequivalence:

δS(t) = φ(t)

q · unit, δN (t) = Q(t)

q(7.78)

where q is an electron’s charge, unit is a parameter in [ Hs ] to ensure proper units for the

components of the electrical circuit and φ(t) is the magnetic flux in [W b]. We also recallstandard relations from electromagnetism:

Q = C · vC (7.79)

iC = dQ

dt(7.80)

φ(t) = L · iL (7.81)

vL = dφ

dt(7.82)

Now, we use (7.78) and relations (7.79)–(7.82) to convert small-signal equations (7.74) and(7.75) into voltage and current relations. After simple algebra, from (7.74) one finds

vL(t) = −�S · L · iL(t) + σN · C · unit · vC (7.83)

This equation can be identified with loop equation for the RLC circuit shown in Fig. 7.23.A loop equation gives

vC = vL + vRp (7.84)

In order for the above equations to be identical, one makes the following identifications:

RS = �S · L = �S · unit

σS, C = 1

unit · σN(7.85)

196 Optical sources

Similarly, from (7.75) one obtains

iC(t) = −�N · C · vC(t) − σS

unit· L · iL(t) (7.86)

That equation can be compared with the node equation from a circuit in Fig. 7.23:

iC = −iR − iL (7.87)

For those equations to be identical, one sets

RN = 1

�N · C, and L = unit

σS(7.88)

One also has the following equivalence:

δN = C · vC

q, and δS = L · iL

q · unit(7.89)

7.5 Problems

1. Perform large-signal analysis for the log gain model described by Eq. (7.27).2. Analyse large-signal response to a Gaussian pulse.3. Based on rate equations for multimode operation, create the small-signal equivalent

circuit model of a semiconductor laser. Draw the corresponding electrical circuit andderive equations which establish links between circuit parameters and rate equations.Hint: Consult the paper by Kibar et al. [41].

4. Construct equivalent circuit model for single quantum well laser with carrier transport(follow Lau’s work in [42]). Introduce carrier density in an SCH region and allow forits interaction with densities in the quantum well.

7.6 Project

1. Conduct multimode analysis of a semiconductor laser. Formulate multimode rate equa-tions. Gain spectrum approximate as described by Coldren and Corzine [4].



197 Appendix 7A



7A.1.1 small signal.m Small-signal analysis7A.1.2 param rate eq bulk.m Parameters for bulk active region in rate eqs.7A.2.1 small epsilon.m Small-signal response with gain compression7A.2.2 param rate eq QW.m Parameters for QW active region in rate eq.7A.3.1 large signal.m Large-signal analysis7A.3.2 eqs large.m Defines rate equations for large-signal analysis

Listing 7A.1.1 Program small signal.m. MATLAB program to perform small-signal anal-ysis for a quantum well semiconductor laser.

% File name: small_signal.m

% Purpose:

% Determines response function

clear all

param_rate_eq_bulk % input data

%

% loop over frequency in GHz


f_min = 0.01; f_max = 100;

freq_GHz = linspace(f_min,f_max,N_max); % From 0.1 to 2 GHz

semilogx(freq_GHz,0);

%

freq = freq_GHz*1d9; % convert frequency to 1/s

omega = 2*pi*freq;

%

hold on

for power_out = [1d-4 1d-3 0.01 0.1 1.0] % values of output power [W]

% Determine steady-state photon density from a given output power

S_zero = 2*power_out/(v_g*alpha_m*h_Planck*freq_ph*V_p);

%

% Construct denominator

D = - omega.^2 + 1j*omega*(1/tau +v_g*a*S_zero) + v_g*a*S_zero/tau_p;

%

response = (v_g*a*S_zero/tau_p)./abs(D);

response_dB = 10*log(response);

%

h = semilogx(freq_GHz,response_dB);

end

xlabel(’frequency [GHz]’,’FontSize’,14); % size of x label

ylabel(’Response function [dB]’,’FontSize’,14); % size of y label

198 Optical sources


axis([f_min f_max -20 35]);

text(0.05, 7, ’10 \muW’,’Fontsize’,14)

text(0.2, 18, ’1 mW’,’Fontsize’,14)

text(0.6, 27, ’10 mW’,’Fontsize’,14)

text(2.5, 33, ’0.1W’,’Fontsize’,14)

text(8, 26, ’1W’,’Fontsize’,14)

pause

close all

Listing 7A.1.2 Function param rate eq bulk.m. Function contains parameters for the rateequation model for the bulk active region.

% File name: param_rate_eq_bulk.m

% Purpose:

% Contains parameters for rate equation model for bulk

% active region

% Source

% G.P. Agrawal and N.K. Dutta

% Long-wavelength semiconductor lasers

% Van Nostrand 1986

% Table 6.1, p. 227

%

% General constants

c = 3d10; % velocity of light [cm/s]

q = 1.6021892d-19; % elementary charge [C]

h_Planck = 6.626176d-34; % Planck constant [J s]

hbar = h_Planck/(2.0*pi); % Dirac constant [J s]

% Geometrical dimensions bulk active region

length = 250d-3; % cavity length [cm]; 250 microns

width = 2d-3; % active region width [cm]; 2 microns

thickness = 0.2d-3; % thickness of an active region [cm]; 0.2 microns

volume_active = length*width*thickness; % volume of active region

%

conf = 0.3; % confinement factor [dimensionless]

V_p = volume_active/conf; % cavity volume [cm^3]

ref_index = 3.4; % effective mode index

%

v_g = c/ref_index; % group velocity [cm/s]

tau_p = 1.6d-12; % photon life-time [s]

tau = 2.2d-9; % carrier life-time [2.71 ns]

%

a = 2.5d-16; % differential gain (linear model) [cm^2]

% Parameters needed to determine output power

alpha_m = 45; % mirror reflectivity [cm^-1]

lambda_ph = 1.3d-3; % laser wavelength [microns]; 1.3 microns

freq_ph = v_g/lambda_ph; % phonon frequency

199 Appendix 7A

Listing 7A.2.1 Function small epsilon.m. Driver program which determines small-signalresponse with gain compression for a single quantum well. For large-signal analysis basedon rate equations.

% File name: small_epsilon.m

% Purpose:

% Determines response function for quantum well with epsilon

clear all

param_rate_eq_QW % input data

%

% loop over frequency in GHz


f_min = 0.1; f_max = 100;

freq_GHz = linspace(f_min,f_max,N_max); % From 0.1 to 2 GHz

semilogx(freq_GHz,0); % make axis and force log scale

%

freq = freq_GHz*1d9; % convert frequency to 1/s

omega = 2*pi*freq;

power_out = 0.001; % output power (10 mW/facet)

%

hold on

for epsilon = [0 1.5d-17 5d-17] % values of epsilon

% Determine steady-state photon density from a given output power

S_zero = 2*power_out/(v_g*alpha_m*h_Planck*freq_ph*V_p);

%

% Construct denominator

A = 1/tau + v_g*S_zero*a/(1+epsilon*S_zero);

B = 1/(conf*tau_p) - (1/(conf*tau_p))*S_zero*epsilon/(1+epsilon*S_zero);

C = conf*v_g*S_zero*a/(1+epsilon*S_zero);

D = (1/tau_p)*S_zero*epsilon/(1+epsilon*S_zero);

damping = A + D;

omega_R2 = A*D + C*B;

H = - omega.^2 + 1j*omega.*damping + omega_R2;

%

response = (A*D+C*B)./abs(H);

response_dB = 10*log(response);

h = plot(freq_GHz,response_dB);

end

xlabel(’frequency [GHz]’,’FontSize’,14); % size of x label

ylabel(’Response function [dB]’,’FontSize’,14); % size of y label


axis([f_min f_max -20 20]);

text(2, 13, ’\epsilon = 0’,’Fontsize’,14)

text(2, 10, ’\epsilon = 1.5 \times 10^{-17}’,’Fontsize’,14)

text(2, 7, ’\epsilon = 5 \times 10^{-17}’,’Fontsize’,14)

pause

close all

200 Optical sources

Listing 7A.2.2 Function param rate eq QW.m. Function which contains rate equationparameters for a single quantum well.

% File name: param_rate_eq_QW.m

% Purpose:

% Contains parameters for rate equation model for QW active region

% Source

% L.A. Coldren and S.W. Corzine,

% "Diode Lasers and Photonic Integrated Circuits", Wiley 1995.

% General constants

c = 3d10; % velocity of light [cm/s]

q = 1.6021892d-19; % elementary charge [C]

h_Planck = 6.626176d-34; % Planck constant [J s]

hbar = h_Planck/(2.0*pi); % Dirac constant [J s]

% Geometrical dimensions

% QW active region

L = 250d-4; % cavity length [cm]; 250 microns

w = 2d-4; % active region width [cm]; 2 microns

d = 80d-8; % thickness of an active region [cm]; 80 Angstroms

%

%V = L*w*d;

V = 4d-12; % volume of active region

%

conf = 0.03; % confinement factor [dimensionless]

V_p = V/conf; % cavity volume [cm^3]

ref_index = 4.2; % effective mode index

%

v_g = c/ref_index; % group velocity [cm/s]

tau_p = 2.77d-12; % photon life-time [s]

tau = 2.71d-9; % carrier life-time [s]

beta_sp = 0.8d-4; % spontaneous emission factor

%

a = 5.34d-16; % differential gain (linear model) [cm^2]

N_tr = 1.8d18; % carrier density at transparency [cm^-3]

eta_i = 0.8;

epsilon = 1d-17;

current_th = 1.11d-3; % current at threshold [A]; 1.11 mA

%

% Parameters needed to determine output power

alpha_m = 45; % mirror reflectivity [cm^-1]

lambda_ph = 1.3d-3; % laser wavelength [microns]; 1.3 microns

freq_ph = v_g/lambda_ph; % phonon frequency

201 Appendix 7A

Listing 7A.3.1 Function large signal.m. Driver program for large-signal analysis basedon rate equations.

% File name: large_signal.m

% Purpose:

% Driver which controls computations for large signal rate equations

% function z = large_signal

%

clear all

tspan = [0 2d-9]; % time interval, up to 2 ns

y0 = [0, 0]; % initial values of N and S

%

[t,y] = ode45(’eqs_large’,tspan,y0);

%

size(t);

t=t*1d9;

y_max = max(y);

y1 = y_max(1);

y2 = y_max(2);

h = plot(t,y(:,1)/y1,’-.’, t, y(:,2)/y2); % divided to normalize


xlabel(’time [ns]’,’FontSize’,14); % size of x label

ylabel(’Arbitrary units’,’FontSize’,14); % size of y label


legend(’carrier density’, ’photon density’) % legend inside the plot

pause

close all

Listing 7A.3.2 Function eqs large.m. Function defines rate equations equations for large-signal analysis.

% File name: eqs_large.m

% Purpose:

% Large signal rate equations are established

% N == y(1)

% S == y(2)

% assumed linear gain model

function ydot = eqs_large(t,y)

%

param_rate_eq_QW % input of needed parameters

%

current = 3d-2; % bias current (step function) [A]; 3 mA

A = v_g*a*(y(1) - N_tr)/(1+epsilon*y(2));

ydot(1) = eta_i*current/(q*V) - y(1)/tau - A*y(2);

ydot(2) = conf*A*y(2) - y(2)/tau_p + conf*beta_sp*y(1)/tau;

ydot = ydot’; % must return column vector

202 Optical sources

References

[1] A. Yariv. Quantum Electronics. Third Edition. Wiley, New York, 1989.[2] C. C. Davis. Lasers and Electro-Optics. Fundamentals and Engineeing. Cambridge

University Press, Cambridge, 2002.[3] B. Mroziewicz, M. Bugajski, and W. Nakwaski. Physics of Semiconductor Lasers.

Polish Scientific Publishers/North-Holland, Warszawa, Amsterdam, 1991.[4] L. A. Coldren and S. W. Corzine. Diode Lasers and Photonic Integrated Circuits.

Wiley, New York, 1995.[5] J. Carroll, J. Whiteaway, and D. Plumb. Distributed Feedback Semiconductor Lasers.

The Institution of Electrical Engineers, London, 1998.[6] E. Kapon, ed. Semiconductor Lasers. Academic Press, San Diego, 1999.[7] D. Sands. Diode Lasers. Institute of Physics Publishing, Bristol and Philadelphia,

2005.[8] C. W. Wilmsen, H. Temkin, and L. A. Coldren, eds. Vertical-Cavity Surface-Emitting

Lasers. Cambridge University Press, Cambridge, 1999.[9] G. P. Agrawal and N. K. Dutta. Semiconductor Lasers. Second Edition. Kluwer

Academic Publishers, Boston/Dordrecht/London, 2000.[10] M. G. A. Bernard and G. Duraffourg. Laser conditions in semiconductors. Physica

Status Solidi, 1:699–703, 1961.[11] G. Keiser. Optical Fiber Communications. Third Edition. McGraw-Hill, Boston,

2000.[12] S.-L. Chuang. Physics of Optoelectronic Devices. Wiley, New York, 1995.[13] J.-M. Liu. Photonic Devices. Cambridge University Press, Cambridge, 2005.[14] G. Lasher and F. Stern. Spontaneous and stimulated recombination radiation in semi-

conductors. Phys. Rev., 133:A553–63, 1964.[15] C. J. Hwang. Properties of spontaneous and stimulated emission in GaAs junction

lasers. I. Densities of states in the active regions. Phys. Rev. B, 2:4117–25, 1970.[16] F. Stern. Calculated spectral dependence of gain in excited GaAs. J. Appl. Phys.,

47:5382–6, 1976.[17] M. S. Wartak, P. Weetman, T. Alajoki, J. Aikio, V. Heikkinen, N. Pikhin, and P. Rusek.

Optical modal gain in multiple quantum well semiconductor lasers. Canadian Journalof Physics, 84:53–66, 2006.

[18] S.-L. Chuang. Physics of Photonic Devices. Wiley, New York, 2009.[19] J. P. Loehr. Physics of Strained Quantum Well Lasers. Kluwer Academic Publishers,

Boston, 1998.[20] P. S. Zory, Jr., ed. Quantum Well Lasers. Academic Press, Boston, 1993.[21] T.-A. Ma, Z.-M. Li, T. Makino, and M. S. Wartak. Approximate optical gain formulas

for 1.55 µm strained quaternary quantum well lasers. IEEE J. Quantum Electron.,31:2934, 1995.

[22] T. Makino. Analytical formulas for the optical gain of quantum wells. IEEE J. Quan-tum Electron., 32:493–501, 1996.

203 References

[23] S. Balle. Simple analytical approximations for the gain and refractive index spectrain quantum-well lasers. Phys. Rev. A, 57:1304–12, 1998.

[24] L. D. Westbrook. Measurements of dg/dN and dn/dN and their dependence on photonenergy in λ = 1.5 µm InGaAsP laser diodes. IEE Proceedings J, 133:135–42, 1986.

[25] L. D. Westbrook. Measurements of dg/dN and dn/dN and their dependence on photonenergy in λ = 1.5 µm InGaAsP laser diodes. Addendum. IEE Proceedings J., 134:122,1987.

[26] H. Ghafouri-Shiraz and B. S. L. Lo. Distributed Feedback Laser Diodes. Wiley,Chichester, 1996.

[27] M.-C. Amann and J. Buus. Tunable Laser Diodes. Artech House, Boston, 1998.[28] M. J. Connelly. Semiconductor Optical Amplifiers. Kluwer Academic Publishers,

Boston, 2002.[29] K. J. Ebeling, ed. Integrated Optoelectronics. Springer-Verlag, Berlin, 1989.[30] R. S. Tucker. High-speed modulation of semiconductor lasers. J. Lightwave Technol.,

3:1180–92, 1985.[31] M. M.-K. Liu. Principles and Applications of Optical Communications. Irwin,

Chicago, 1996.[32] M. Ahmed and A. El-Lafi. Analysis of small-signal intensity modulation of semicon-

ductor lasers taking account of gain suppression. PRAMANA, 71:99–115, 2008.[33] R. S. Tucker. High-speed modulation of semiconductor lasers. IEEE Trans. Electron

Devices, 32:2572–84, 1985.[34] A. Yariv and P. Yeh. Photonics. Optical Electronics in Modern Communications. Sixth

Edition. Oxford University Press, New York, Oxford, 2007.[35] K. Otsuka and S. Tarucha. Theoretical studies on injection locking and injection-

induced modulation of lase diodes. IEEE J. Quantum Electron., 17:1515, 1981.[36] D. Marcuse and T.-P. Lee. On approximate analytical solutions of rate equations for

studying transient spectra of injection lasers. IEEE J. Quantum Electron., 19:1397,1983.

[37] M. S. Demokan and A. Nacaroglu. An analysis of gain-switched semiconductorlasers generating pulse-code-modulated light with a high bit rate. IEEE J. QuantumElectron., 20:1016, 1984.

[38] J. Katz, S. Margalit, C. Harder, D. Wilt, and A. Yariv. The intrinsic electrical equivalentcircuit of a laser diode. IEEE J. Quantum Electron., 17:4–7, 1981.

[39] C. Harder, J. Katz, S. Margalit, J. Shacham, and A. Yariv. Noise equivalent circuit ofa semiconductor laser diode. IEEE J. Quantum Electron., 18:333–7, 1982.

[40] R. S. Tucker and D. J. Pope. Circuit modeling of the effect of diffusion on damping ina narrow-stripe semiconductor laser. IEEE J. Quantum Electron., 19:1179–83, 1983.

[41] O. Kibar, D. Van Blerkom, C. Fan, P. J. Marchand, and S. C. Esener. Small-signal-equivalent circuits for a semiconductor laser. Applied Optics, 37:6136–9, 1998.

[42] K. Y. Lau. Dynamics of quantum well lasers. In P. S. Zory, Jr, ed., Quantum WellLasers, pages 217–75. Academic Press, Boston, 1993.

[43] E. J. Flynn. A note on the semiconductor laser equivalent circuit. J. Appl. Phys.,85:2041–5, 1999.

8 Optical amplifiers and EDFA

As signals travel in optical communication systems, they are attenuated by optical fibre.Eventually, after some distance they can become too weak to be detected. One possible wayto avoid this situation is to use optical amplifiers to increase the amplitude of the signal.An optical amplifier is an optical device for amplifying signals propagating over severalchannels in optical fibre to compensate for their loss during propagation. Signals also getdistorted, say by presence of noise, and the amplification process does not clean them norreshape them. The cleaning process is commonly known as regeneration and will not bediscussed here.

In this chapter we will concentrate on optical amplifiers. We review their general prop-erties and will also discuss erbium doped fibre amplifiers (EDFA). Semiconductor opticalamplifiers (SOA), which are essentially semiconductor lasers without mirrors so they donot lase, will be discussed in the next chapter.

Possible applications of optical amplifiers are illustrated in Fig. 8.1. Some of the basicapplications are listed below:

a) as in-line amplifiers for power boosting,b) as pre-amplifiers to increase the received power at the receiver,c) as power amplifiers to increase transmitted power,d) as a power booster in a local area network.

The main types of optical amplifiers are:

• doped fibre amplifiers• Raman and Brillouin amplifiers, and• semiconductor optical amplifiers.

Issues which will be discussed in relation to those amplifiers are: gain and gain bandwidth,gain saturation, and noise and noise figure.

In short, the operation and characteristics of all types of amplifiers are:

1. population inversion is created, which means that more systems (atoms, molecules) arein a high energy state than in a lower one,

2. the incoming pulses of signal induce stimulated emission,3. amplifiers saturate above a certain signal power,4. amplifiers add noise to the signal.

In this book we will only describe two types of optical amplifiers: semiconductor opticalamplifiers (SOA) and erbium doped fibre amplifiers (EDFA). Their general characteris-tics are compared in Table 8.1 where we summarize essential properties of EDFA and

204

205 General properties

Table 8.1 Some properties of EDFA and SOA.

Properties of amplifiers EDFA SOA

Active medium Er3+ ion in silica Electron-holein semiconductors

Typical length Few metres 500 µmPumping Optical ElectricalGain spectrum 1.5–1.6 µm 1.3–1.5 µmGain bandwidth 24–35 nm 100 nmRelaxation time 0.1–1 ms <10–100 psMaximum gain 3–50 dB 25–30 dBSaturation power >10 dBm 0–10 dBmCrosstalk – For bit rate 10 GHzPolarization Insensitive SensitiveNoise figure 3–4 dB 6–8 dBInsertion loss <1 dB 4–6 dBOptics Pump laser diode couplers, Antireflection coatings,

fibre splice fibre-waveguide couplingIntegration No Yes

Optical fibre Optical fibreIn-lineamplifier

TX RX

Optical fibreTX RX

Fig. 8.1 Possible applications of optical amplifiers: as an in-line amplifier (top) or as a preamplifier (bottom). TX stands fortransmitter, RX for receiver.

SOA, after Yariv and Yeh [1] (reproduced with permission). Next, we will discuss generalcharacteristics of optical amplifiers.

Some books on the subject are: Desurvire [2], Becker et al. [3] and Bjarklev [4].

8.1 General properties

Amplification in optical amplifiers is through stimulated emission (the same as in lasers).One can therefore consider similar characteristic parameters, like gain and its spectrum,bandwidth, etc. We will discuss them in some detail.


8.1.1 Gain spectrum and bandwidth

To model gain, one typically starts with a homogeneously broadened two-level system.Local gain coefficient for such system is [5]

g (ν, P) = g0

1+ (ν−ν0)2

�ν20

+ PPsat

(8.1)

where g0 is the peak value of the unsaturated gain, ν0 atomic transition frequency, �ν0 3-dBlocal gain bandwidth, Psat saturation power and P and ν are optical power and frequency ofthe amplified signal.

Local gain can also be written as

g (ω, P) = g0

1 + (ω − ω0)2 T 2

2 + P/Psat

(8.2)

where ω = 2πν is the angular frequency and T2 is known as dipole relaxation time [6].For small signal power one can consider a unsaturated regime defined as P � Psat . In

this limit local gain is

g (ω) = g0

1 + (ω − ω0)2 T 2

2

(8.3)

The following conclusions can be derived from the above equation:

1. maximum gain corresponds to transitions with angular frequency ω = ω0,2. for ω �= ω0 gain spectrum is described by Lorentzian profile,3. local gain bandwidth, which is defined as the full width at half maximum (FWHM), is

�ω0 = 2

T2(8.4)

Local gain bandwidth �ω0 is defined by points in frequency where local gain takes half thevalue at the maximum. In terms of frequency it can be written as

�ν0 = �ω0

2π= 1

πT2(8.5)

Let P (z) be the optical power at a distance z from the input end. Its change is describedas

dP(z)

dz= g(ν, P) · P(z) (8.6)

Assume a linear device (here for power levels P � Psat ) where local gain is independent ofthe signal power. Integration of the above equation gives

P (z) = P(0) eg·z (8.7)

where P(0) = Pin is the signal input power. Linear amplifier gain is defined as

G = P(L)

P(0)(8.8)

207 General properties

where P(L) = Pout is the output power. From the above solution one obtains

G = P (L)

P(0)= egL = exp

⎡⎣ g0 · L

1 + (ν−ν0 )2

�ν20

⎤⎦ (8.9)

Amplifier bandwidth B0 is evaluated using the above solution. It is defined by two frequencypoints where power drops by 50%, i.e. P3dB = 1

2 Pmax(L), which translates into G3dB =12 Gmax. Here Gmax is the maximum value of gain evaluated at ν = ν0. In detail

exp

⎡⎣ g0 · L

1 + B20

�ν20

⎤⎦ = 1

2exp

⎡⎣ g0 · L

1 + 0�ν2

0

⎤⎦ = 1

2exp (g0 · L)

where we have introduced 3 − dB bandwidth B0 as B0 = ν − ν0. By the straightforwardalgebra, from the above relation one finds

B0 = �ν0

√ln 2

g0L − ln 2(8.10)

Macroscopic bandwidth of the amplifier B0 is smaller than the local gain bandwidth �ν0.

8.1.2 Gain saturation

We will now analyse the gain when signal power has large value and saturation effects arebecoming important. Assume that ω = ω0. Substituting Eq. (8.2) into (8.6), one obtains

dP

dz= g0P

1 + P/Psat(8.11)

We introduce new variable u = P/Psat and use separation of variables method to integratethe above equation. One obtains∫ uout

uin

1 + u

udu =

∫ L

0g0 dz

where uin = Pin/Psat , uout = Pout/Psat and Pin, Pout are input and output powers, respectively.L is the length of an amplifier. Gain G is defined as

G = Pout

Pin(8.12)

One obtains

G = G0 exp

(−G − 1

G

Pout

PS

)(8.13)

where G0 = exp (g0 · L).In Fig. 8.2 we have plotted saturation gain dependence from Eq. (8.13). MATLAB code

and its description are provided in Appendix, Listings 8A.1.1, 8A.1.2 and 8A.1.3.


10−2 10−1 100 1010

0.2

0.4

0.6

0.8

1

Normalized output power, Pout/Psat

Nor

mal

ized

am

plif

ier

gain

, G/G

0

G0 = 10

G0 = 100

G0 = 1000

Fig. 8.2 Saturated normalized amplifier gainG/G0 as a function of the normalized output power for three values of theunsaturated amplifier gainG0.

8.1.3 Amplifier noise

Signal-to-noise ratio (SNR) of optical amplifiers is degraded because spontaneous emissionadds to the signal during its amplification. Amplifier noise figure Fn is defined as [7]

Fn = (SNR)in

(SNR)out(8.14)

SNR refers to the electrical power generated when the signal is converted to electricalcurrent by using a photodetector. We model Fn by considering an ideal detector limited onlyby a shot noise

(SNR)in = 〈I〉2

σ 2s

(8.15)

where 〈I〉 = RPin is the average photocurrent, R = qhν

is the responsivity of an ideal detectorwith unit quantum efficiency, σ 2

s = 2q (RPin) � f is the variance from shot noise and � f isthe detector bandwidth. At the output, we should add spontaneous emission to the receivernoise

Ssp (ν) = (G − 1) nsphν (8.16)

Here Ssp is the spectral density of the noise induced by spontaneous emission, ν is opticalfrequency and nsp is the spontaneous-emission factor or population inversion factor. Thevalue of nsp is nsp = 1 for amplifiers with complete population inversion (all atoms in theupper state) and nsp > 1 for incomplete population inversion.

For a two-level system

nsp = N2

N2 − N1(8.17)

209 Erbium-doped fibre amplifiers (EDFA)

Wavelength (nm)

Lo

ss o

r gai

n (d

B/m

)

1480 1500 1520 1540 1560

2 Gain

Absorption4

6

0

8

10

Fig. 8.3 Absorption and gain spectra of Ge silicate amplifier fibre. Copyright (1991) IEEE. Reprinted, with permission from C. R.Giles and E. Desurvire, J. Lightwave Technol., 9, 271 (1991).

where N1 and N2 are the atomic populations in the lower and upper states, respectively.Total variance of the shot noise plus spontaneous emission noise is thus

σ 2 = 2q (RGPin) � f + 4 (GRPin)(RSsp

)� f (8.18)

All other contributions to the receiver noise are neglected. At the output, the SNR of theamplified signal is

(SNR)out = 〈I〉2

σ 2= (RGPin)

2

σ 2≈ GPin

4Ssp� f(8.19)

assuming G � 1 and we neglected first term in (8.18).Using definition of Fn, one finds

Fn = 2nspG − 1

G≈ 2nsp (8.20)

It shows that even for an ideal amplifier (nsp = 1), amplified signal is degraded by a factorof 2 (3 dB). In practice, Fn is in the range 6–8 dB.

8.2 Erbium-doped fibre amplifiers (EDFA)

Experimental data on absorption and gain of an EDFA whose core was codoped with Geto increase the refractive index are shown in Fig. 8.3, after Giles and Desurvire [8].

An optical fibre amplifier consists of an optical fibre where amplification takes placewhich is doped with a rare-earth element, and a pumping light supply system for supplyingpumping light to the optical fibre for amplification, see Fig. 8.4. The pumping light supplysystem usually includes a semiconductor laser and an optical coupler for guiding thepumping light into the optical fibre for amplification.


Pump

Isolator

Wavelength-selectivecouplers

Signal outSignal in

980 nm

1550 nm

Erbium fibre

Fig. 8.4 Schematic of EDFA.

980 nm 1530 nm1530 nm

Non-radiative

(a) (c)(b)

Fig. 8.5 Illustration of amplification in EDFA using a three-level model.

1

2

3

32

21

pp

ss

σ

σ φ

φ Γ

Γ

Fig. 8.6 Illustration of notation for a three-level system.

Erbium-doped amplifiers are made by doping a segment of the fibre with erbium and thenexciting the erbium atoms to a high energy level through the introduction of pumping light.The energy is transferred gradually to signal light passing through the fibre segment duringexcitation, resulting in an amplification of the signal light upon exit from the amplifier.Fibre optic amplifiers can amplify signal light including one or more wavelengths within apredetermined wavelength band without converting them into an electrical signal.

Illustration of the amplification process in EDFA based on the three-level model isshown in Fig. 8.5 for the case of a 980 nm pump. A pump photon at a 980 nm wavelengthis absorbed by an erbium ion in the ground state and jumps into the highest energy level.Then, through a non-radiative decay the ion loses its energy and arrives into a metastablestate. Once there, a photon having wavelength of 1530 nm can force a stimulated transitionof the ion into its ground state, creating one additional photon and an amplification.

In this section, we will describe EDFA using a three-level model. We follow Becker[3]. The model is shown in Fig. 8.6. Level 1 is a ground state, level 2 is known as a

211 Erbium-doped fibre amplifiers (EDFA)

metastable one (it has a long lifetime) and level 3 is an intermediate state. The populationof levels are introduced as N1, N2, N3. The spontaneous transition rates of the ion (transitionprobabilities) which include radiative and also non-radiative contributions are denoted as�32 and �21 and correspond to transitions between levels 3 → 2 and 2 → 1, respectively.σp is the pump absorption cross section and σs is the signal emission cross section. Theincident light intensity fluxes of pump and signal are denoted by φp and φs, respectively.They are defined as number of photons per unit time per unit area.

This three-level model represents energy level structure of Er+ which is involved in theamplification process. To obtain amplification, we need a population inversion betweenlevels 1 and 2. Here, we only consider a one-dimensional model where we assume that thepump and signal intensities and also distribution of Er ions are constant in the transversedirection.

Based on the above observations, the rate equations for the changes of populations forall levels are postulated as

dN1

dt= �21N2 − (N1 − N3) σpφp + (N2 − N1) σsφs (8.21)

dN2

dt= −�21N2 + �32N3 − (N2 − N1) σsφs (8.22)

dN3

dt= −�32N3 + (N1 − N3) σpφp (8.23)

8.2.1 Steady-state analysis

In the steady-state conditions

dN1

dt= dN2

dt= dN3

dt= 0 (8.24)

Also, total population N is assumed to be constant:

N = N1 + N2 + N3 (8.25)

From Eq. (8.23) one obtains

N3 = N11

1 + �32σpφp

(8.26)

In what follows we will assume fast decay from level 3 to level 2; i.e. decay from level3 is dominant compared to pump rate. Mathematically, the assumption corresponds to thefollowing condition �32 � σpφp. Lifetime of level 3, τ32 is related to transition probabilityas τ32 = 1/�32. In this limit there is almost no population of level 3, and therefore N3 ≈ 0.With those assumptions the system can be effectively considered as consisting of two levelsonly which are described by the following equations:

σpφp + σsφs

�21 + σsφsN1 − N2 = 0 (8.27)

N1 + N2 = N (8.28)


Solutions by standard methods give for population inversion

N2 − N1 = Nσpφp − �21

�21 + 2σsφs + σpφp(8.29)

8.2.2 Effective two-level approach

Keeping the above assumption, i.e. �32 � σpφp which allows us to neglect level 3, fromEq. (8.28) one has

dN1

dt= −dN2

dt

It is therefore enough to consider only one equation, say for N2; the other population N1

can be found from N1 = N − N2. Following Pedersen et al. [9], to describe the system wepostulate the following equations:

dN2

dt= −�21N2 + [

σ (a)s N1 − σ (e)

s N2]φs −

[σ (e)

p N2 − σ (a)p N1

]φp (8.30)

dN1

dt= �21N2 + [

σ (e)s N2 − σ (a)

s N1]φs −

[σ (a)

p N1 − σ (e)p N2

]φp (8.31)

where σ (a)s , σ (e)

s , σ (a)p , σ (e)

p represent signal and pump cross sections for absorption andemission, respectively. Assume steady-state and from Eq. (8.30) determine N2:

N2 = Nσ (a)

s φs + σ (a)p φp

1τ

+[σ

(a)s + σ

(e)s

]φs +

[σ

(a)p + σ

(e)p

]φp

where we have introduced τ = 1/�21. Further, introduce signal Is and pump Ip intensitiesas

φs = Is

hνs, φp = Ip

hνp

where h is the Planck constant and νs, νp are frequencies of the signal and pump, respectively.This allows us to write

N2 = Nτ

σ (a)s

hνsIs(z) + τ

σ (a)p

hνpIp(z)

τ σ(a)s +σ

(e)s

hνsIs(z) + τ

σ(a)p +σ

(e)p

hνpIp(z) + 1

We further assume that N is independent of distance along fibre z. The variation of signaland pump intensities is described as

dIs(z)

dz= [

σ (e)s N2 − σ (a)

s N1]

Is(z)

dIp(z)

dz=[σ (e)

p N2 − σ (a)p N1

]Ip(z)

213 Gain characteristics of erbium-doped fibre amplifiers

Table 8.2 Typical parameters of a low-noise in-line EDFA.

Parameter Symbol Value Unit

Signal mode area πw2 1.3 × 10−11 m2

Erbium concentration Ntot 5.4 × 1024 m−3

Signal overlapping integral �s 0.4Pump overlapping integral �p 0.4Signal emission cross section sse 5.3 × 10−25 m2

Signal absorption cross section ssa 3.5 × 10−25 m2

Pump absorption cross section sp 3.2 × 10−25 m2

Signal local saturation power Pss 1.3 mWPump local saturation power Psp 1.6 mW

8.3 Gain characteristics of erbium-doped fibre amplifiers

We outline a basic approach to evaluate the performance of EDFA. Neglecting amplifiedspontaneous emission (ASE) and assuming copropagation configuration, the equationsdescribing steady-state are

dIs(z)

dz= 2π�s

{σseNme(z)Is(z) − σsaNgr(z)Is(z)

}(8.32)

dIp(z)

dz= 2π�pσsaNgr(z)Ip(z) (8.33)

Nme(z) = NtotIs(z)/Iss + Ip(z)/Isp

1 + Ip(z)/Isp + 2Is(z)/Iss(8.34)

Nme(z) + Ngr(z) = Ntot (8.35)

In the previous equations Ngr(z) is the population of the ground state, Nme(z) is the pop-ulation of the metastable state, Ip(z) is the intensity of the pump wave propagating at thewavelength λp and Is(z) is the intensity of the signal wave propagating at the wavelengthλs in the positive z-direction.

Finally, the overall amplifier gain G is obtained using the following relation:

G = Is(L)

Is(0)(8.36)

where L is the doped fibre length.

8.3.1 Typical EDFA characteristics

The above equations were solved numerically using parameter values from Table 8.2 [8],[10], [11]. MATLAB codes are provided in Listings 8A.2.1–8A.4.2.

The variation of gain with fibre length was determined first and the results are summarizedin Fig. 8.7 for different values of pump power. MATLAB code is shown in the Appendix,Listings 8A.2.1–8A.2.4. Constant signal input power of 10µW and constant erbium doping


0 1 2 3 40

2

4

6

8

10

Doped fibre length (m)

Gai

n (a

.u.)

3 mW5 mW7 mW9 mW

Fig. 8.7 Variation of gain versus fibre length for four different values of pump power.

2 3 4 5 6 7 80

5

10

15

20

Pump power (mW)

Gai

n (a

.u.)

5 m10 m15 m

Fig. 8.8 Variation of gain versus pump power for three different values of fibre length.

density have been assumed. Gain was evaluated for four different pump power levels equalto 3, 5, 7 and 9 mW. As it is shown, gain increases up to a certain fibre’s length and thenbegins to decrease after reaching a maximum point. Similar results were reported in theliterature, compare [10] and [11].

The optimum fibre length (the one which corresponds to a maximum gain) is a few metresand it increases with the pump power. The reason for the decrease in gain is insufficientpopulation inversion due to excessive pump depletion.

In Fig. 8.8 we show the variation of gain with pump power for three different values offibre lengths of 5, 10 and 15 m and a constant signal input power equal to 1 mW. MATLAB

215 Appendix 8A

code is shown in the Appendix, Listings 8A.3.1 and 8A.3.2. Constant Er doping densitywas also assumed. As it is seen, gain of the EDFA increases with the increasing pumppower and then saturates after a certain level of pump power. The pump saturation effectoccurs for input powers in the range 3–6 mW. Consult literature for more discussion of theobtained results [10] and [11].

8.4 Problems

1. An EDFA has 20 dB of gain. If the power of incident signal is 150 µW, what is theoutput power?

2. An optical amplifier amplifies an incident signal of 1 µW to 1 mW. Assuming thesaturation power of 20 mW, what will be the output power of 100 µW incident signalon the same amplifier?

3. An optical amplifier produces power gain of 1000 over 30 m with a 10 mW pump.(a) What is the gain exponential coefficient α in nepers per metre? (b) What shouldbe the length of that amplifier to produce gain of 1500?

4. Using steady-state amplifier equations and appropriate approximations, derivean analytical expression for an optimum length of fibre which gives maximumamplification.

8.5 Projects

1. Perform a literature survey on EDFA with noise. Write a MATLAB program to simulateEDFA with realistic noise.

2. Perform a literature survey on Raman amplifiers. Write MATLAB code to simulateRaman amplifier. Compare Raman amplifier and EDFA.

3. Formulate transient model of EDFA with the saturation effect of the forward and back-ward amplified spontaneous emission [12]. Integrate numerically the resulting equations.Analyse gain saturation and recovery times of an EDFA and the effects on the amplifi-cation of optical pulses.






8A.1.1 sat gain.m Creates data for saturated gain8A.1.2 f sat gain.m Function used by sat gain.m8A.1.3 sat plot.m Function plots saturated gain8A.2.1 gain variable length.m Generates date for plots gain vs fibre

length for fixed gain8A.2.2 edf a param.m Parameters for EDFA8A.2.3 edf a eqs.m Equations for EDFA8A.2.4 variable length plot.m Plots gain using data generated by

gain variable length.m8A.3.1 gain variable pump.m Generates data for plots gain vs power

for fixed fibre length8A.3.2 variable gain plot.m Plots gain using data generated by

gain variable pump.m

Listing 8A.1.1 Function sat gain.m. MATLAB program which generates data for satu-rated normalized amplifier gain G0. To generate the required data, it must be run three timesfor three values of G0. Therefore one should appropriately comment inputs for G0. Eachtime a different output file must be specified; i.e. the output file name should be changed.

% File name: sat_gain.m

% Generates data for plots of saturated gain G as a function of

% the output power for three values of the unsaturated amplifier gain G_0

% File must be run three times for three values of G_0

clear all

%

G_0 = 10;

%G_0 = 100;

%G_0 = 1000;

u_init = 1d-2;

Delta_u = 0.5d-1;

u_final = 1d1;

uu = u_init;

yy = fzero(@(y) f_sat_gain(y, u_init, G_0), 0.5);

for u = (u_init + Delta_u):Delta_u:u_final

uu = [uu, u];

y = fzero(@(y) f_sat_gain(y, u, G_0), 0.5);

yy = [yy, y];

end

%

d_u = length(uu);

fid = fopen(’gain_data_10.dat’, ’wt’); % Open the file.

for ii = 1:d_u

217 Appendix 8A

fprintf (fid, ’ %11.4f %11.4f\n’, uu(ii),yy(ii)); % Print the data

end

status = fclose(fid); % Close the file.

Listing 8A.1.2 Function f sat gain.m. MATLAB function used by a program which plotssaturated gain.

% defines function used in plot of saturated gain of optical amplifier

function fun = f_sat_gain(y, u, G_0)

fun = y - exp(-(G_0*y-1)*u/(G_0*y));

Listing 8A.1.3 Function sat plot.m. MATLAB function used to plot saturated gain fromfiles. It was not possible here to create one MATLAB program which will perform all thecalculations and also to plot all the results. Instead, we had to run sat gain.m three timesfor three different values of G0, output the results to data files and then plot final resultsusing a separate program.

% File_name: sat_plot.m

% Plots graph of saturated gain using data generated by sat_gain.m

clear all

fid = fopen(’gain_data_10.dat’);

a_10 = fscanf(fid,’%f %f’,[2 inf]); % It has two rows

fclose(fid);

%



fclose(fid);

%



fclose(fid);

%

semilogx(a_10(1,:),a_10(2,:),’k-’, a_100(1,:),a_100(2,:),’k--’,...

a_1000(1,:),a_1000(2,:),’k-.’,’LineWidth’,1.5)

%

xlabel(’Normalized output power, P_{out}/P_{sat}’,’FontSize’,14);

ylabel(’Normalized amplifier gain, G/G_0’,’FontSize’,14);

legend(’G_0 = 10’,’G_0 = 100’,’G_0 = 1000’)


pause

close all


Listing 8A.2.1 Function gain variable length.m. MATLAB function used to computegain versus fibre length for a fixed value of pump power.

% File name: gain_variable_length.m

% Computation of gain vs fibre length for fixed value of pump power

% User selects pump power which is controlled by P_p

% Calculations are repeated for P_p = 3,5,7,9 d-3 Watts

% User should also appropriately rename output file ’gain_P_3.dat’

% based on Iannone-book, p.86

% P_s == y(1) signal power

% P_p == y(2) pump power

%

clear all

gain = 0.0;

gain_temp = 0.0;

P_s = 100d-7; % signal power [Watts]

P_p = 9d-3; % CHANGE % pump power [Watts]

%

for d_L = 0.01:0.01:10

span = [0 d_L];

y0 = [P_s P_p]; % initial values of [P_s P_p]

[z,y] = ode45(’edfa_eqs’,span,y0); % z - distance

len = length(z);

gain_temp = y(len)/y(1);

gain = [gain, gain_temp];

end

gain_log = log(gain);

d_L_plot = 0.01:0.01:10;

d_L_plot = [0, d_L_plot];

% plot(d_L_plot, gain_log) % uncomment if you want to see plot when run

% axis([0 4 0 5])

% pause

% close all

%

d_u = length(d_L_plot);

fid = fopen(’gain_P_9.dat’, ’wt’); % CHANGE % Open the file.

for ii = 1:d_u

fprintf (fid, ’ %11.4f %11.4f\n’, d_L_plot(ii),gain_log(ii));

end

status = fclose(fid); % Close the file

219 Appendix 8A

Listing 8A.2.2 Function edfa param.m. MATLAB function contains parameters for amodel of EDFA.

% File name: edfa_param.m

%-----------------------------------------------------------------

% Purpose:

% Contains parameters for model of EDFA based on Table 3.2

% Iannone-book

N_tot = 5.4d24; % Erbium concentration (m^-3)

Gamma_s = 0.4; % Signal overlapping integral (dimensionless)

Gamma_p = 0.4; % Pump overlapping integral (dimensionless)

s_se = 5.3d-25; % Signal emission cross-section (m^2)

s_sa = 3.5d-25; % Signal absorption cross-section (m^2)

s_p = 3.2d-25; % Pump absorption cross-section

P_ss = 1.3d-3; % Signal local saturation power (W)

P_sp = 1.6d-3; % Pump local saturation power (W)

Listing 8A.2.3 Function edfa eqs.m. MATLAB function contains equations for EDFA.

function y_z = edfa_eqs(z,y)

% Purpose:

% To establish equations for EDFA following Iannone-book, p.86



%

edfa_param % input of needed parameters

num = y(1)/P_ss + y(2)/P_sp;

denom = y(2)/P_sp + 2*y(1)/P_ss + 1.0;

N_me = N_tot*num/denom;

N_gr = N_tot - N_me;

%

y_z(1) = 2*pi*Gamma_s*(s_se*N_me*y(1) - s_sa*N_gr*y(1)); % derivative

y_z(2) = -2*pi*Gamma_p*s_p*N_gr*y(2);

y_z = y_z’; % must return column vector

end

Listing 8A.2.4 Function variable length plot.m. MATLAB function used to plot graphof gain using data generated by gain variable length.m.

% File_name: variable_length_plot.m

% Plots graph of gain using data generated by ’gain_variable_length.m’

clear all

% Open files for 4 values of length of device: 5 m,10 m,15 m

fid = fopen(’gain_P_3.dat’);



fclose(fid);

%



fclose(fid);

%



fclose(fid);

%



fclose(fid);

%

plot(a_3(1,:),a_3(2,:),’k-’, a_5(1,:),a_5(2,:),’k--’,...

a_7(1,:),a_7(2,:),’k-.’,a_9(1,:),a_9(2,:),’k-’,’LineWidth’,1.5)

axis([0 4 0 10])

xlabel(’Doped fiber length (m)’,’FontSize’,14);

ylabel(’Gain (a.u)’,’FontSize’,14);

legend(’3 mW’,’5 mW’,’7 mW’,’9 mW’)


pause

close all

Listing 8A.3.1 Function gain variable pump.m. MATLAB function used to compute gainversus pump power for fixed value of fibre length.

% File name: gain_variable_pump.m

% Computation of gain vs power pump for fixed value of fiber length

% User selects fiber length which is controlled by d_L

% Calculations are repeated for d_L = 5,10,15 meters

% User should also appropriately rename output file ’gain_L_5.dat’

% Based on Iannone-book, p.86



%

clear all

gain = 0.0;

gain_temp = 0.0;

P_s = 100d-5; % signal power [Watts]

d_L = 5; % CHANGE % fiber length [meters]

span = [0 d_L]; % region of integration [meters]

for P_p = 0.1d-3:0.1d-3:10d-3 % pump power [Watts]

y0 = [P_s P_p]; % initial values of [P_s P_p]

[z,y] = ode45(’edfa_eqs’,span,y0); % z - distance

len = length(z);

221 Appendix 8A

gain_temp = y(len)/y(1);

gain = [gain, gain_temp];

end

gain_log = log(gain);

P_p_plot = 0.1:0.1:10;

P_p_plot = [0, P_p_plot];

% plot(P_p_plot, gain_log) % uncomment if you want to see plot when run

% axis([0 10 0 30])

% pause

% close all

%

d_u = length(P_p_plot);

fid = fopen(’gain_L_5.dat’, ’wt’); % CHANGE % Open the file.

for ii = 1:d_u

fprintf (fid, ’ %11.4f %11.4f\n’, P_p_plot(ii),gain_log(ii));

end

status = fclose(fid); % Close the file

Listing 8A.3.2 Function variable pump plot.m. MATLAB function used to plot graph ofgain using data generated by gain variable pump.m.

% File_name: variable_pump_plot.m

% Plots graph of gain using data generated by ’gain_variable_pump.m’

clear all

% Open files for 4 values of length of device: 5 m,10 m,15 m

fid = fopen(’gain_L_5.dat’);


fclose(fid);

%



fclose(fid);

%



fclose(fid);

%

plot(a_5(1,:),a_5(2,:),’k-’, a_10(1,:),a_10(2,:),’k--’,...

a_15(1,:),a_15(2,:),’k-.’,’LineWidth’,1.5)

axis([2 8 0 20])

xlabel(’Pump power (mW)’,’FontSize’,14);

ylabel(’Gain (a.u)’,’FontSize’,14);

legend(’5 m’,’10 m’,’15 m’)


pause

close all


References

[1] A. Yariv and P. Yeh. Photonics. Optical Electronics in Modern Communications. SixthEdition. Oxford University Press, New York, Oxford, 2007.

[2] E. Desurvire. Erbium-Doped Fiber Amplifiers, Principles and Applications. Wiley,New York, 2002.

[3] P. C. Becker, N. A. Olsson, and J. R. Simpson. Erbium-Doped Fiber Amplifiers.Fundamentals and Technology. Academic Press, San Diego, 1999.

[4] A. Bjarklev. Optical Fiber Amplifiers: Design and System Applications. Artech House,Boston, 1993.

[5] A. Yariv. Quantum Electronics. Third Edition. Wiley, New York, 1989.[6] G. P. Agrawal. Fiber-Optic Communication Systems. Second Edition. Wiley, New

York, 1997.[7] G. P. Agrawal. Semiconductor optical amplifiers. In G. P. Agrawal, ed., Semiconductor

Lasers. Past, Present, and Future, pages 243–83. American Institute of Physics Press,Woodbury, New York, 1995.

[8] C. R. Giles and E. Desurvire. Modeling erbium-doped fiber amplifiers. J. LightwaveTechnol., 9:271–83, 1991.

[9] B. Pedersen, A. Bjarklev, O. Lumholt, and J. H. Povlsen. Detailed design analysis oferbium-doped fiber amplifiers. IEEE Photon. Technol. Lett., 3:548–50, 1991.


[11] R. Sabella and P. Lugli. High Speed Optical Communications. Kluwer AcademicPublishers, Dordrecht, 1999.

[12] K. Y. Ko, M. S. Demokan, and H. Y. Tam. Transient analysis of erbium-doped fiberamplifiers. IEEE Photon. Technol. Lett., 6:1436–8, 1994.

9 Semiconductor optical amplifiers (SOA)

The field of semiconductor optical amplifiers (SOA) is one of the fast growing in recentyears. Many new applications of SOA were proposed. For comprehensive summaries, seebooks by Connolly [1] and Dutta and Wang [2].

In this chapter we will discuss the following topics:

• general concepts• amplifier equations• influence of cavity (FP amplifiers)• gain dependence on polarization and temperature• some applications.

Typical use of SOA in an amplifier configuration is shown in Fig. 9.1. Here, a signalfrom transmission fibre is imputed on SOA where it is amplified. After amplification it isredirected again to the fibre. Coupling between fibre and SOA is provided by an appropriateoptical element.

9.1 General discussion

SOA is very similar to a semiconductor laser. There are two categories of SOA (see Fig. 9.2):(a) Fabry-Perot (FP) amplifier and (b) travelling-wave amplifier (TWA). The FP amplifierdisplays high gain but has a non-uniform gain spectrum, whereas TWA has broadband gainbut requires very low facet reflectivities. The FP amplifier has large reflectivities at bothends which results in resonant amplification, and also has large gain at the wavelengthcorresponding to longitudinal modes of the FP cavity.

TWA has very small reflectivities, achieved by AR (anti-reflection) coating; its gainspectrum is broad but small ripples exist in gain spectrum, resulting from residual facetreflectivity. It is more suitable for system applications but the gain must be polarizationindependent. The phenomenological expression for gain of SOA is written as

gm = a(n − n0) − a2(λ − λp)2 (9.1)

and the wavelength peak value as

λp = λ0 + a3(n − n0) (9.2)

Typical values of the parameters which appear in the above formulas are given inTable 9.1, from [3].

223


Table 9.1 Basic parameters of SOA.

Description Symbol Value

Differential gain a 2.7 × 10−16 cm2

Gain coefficient a2 0.15 cm−1 nm−2

Gain coefficient a3 2.7 × 10−17 nm · cm−3

Transparency density n0 1.1 × 1018 cm−3

Amplifier length L 350 µmDensity at threshold nth 1.8 × 1018 cm−3

core

cladding

Transmission fibre Transmission fibre

Injection current (pump)

Active region

coupling optics

SOA

Opticalinputsignal

Amplifiedopticalsignal

Fig. 9.1 SOA in an amplifier configuration.

Balksemiconductor

Injection current (pump) Injection current (pump)

Active regionActive region

Opticalinputsignal

Cleavedfacet

(a)

Cleavedfacet


Balksemiconductor

Opticalinputsignal

(b)


Fig. 9.2 Types of SOA: Fabry-Perot (left) and travelling-wave (right).

From the previous equations, the 3 dB gain bandwidth is determined as

2�λ = 2

√a(n − n0)

2n2(9.3)

Substituting typical values from Table 9.1 gives SOA bandwidth equal to 54 nm.

225 General discussion

Ein EoutE1E2E3

R2R1

L

Fig. 9.3 Basic model of a Fabry-Perot amplifier.

9.1.1 Gain formula for SOA with facet reflectivities

Consider a typical Fabry-Perot ethalon with a gain medium in-between, see Fig. 9.3. Totaloutput electric field Eout consists of all transmitted contributions:

Eout = E1 + E2 + E3 + · · ·Here R1 and R2 are coefficients of internal reflections for electric field, and 1 − R1 and1 − R2 corresponding coefficients of transmission. The single-pass gain Gs for field is

Gs = e�(g−α)·L (9.4)

where � is optical confinement factor, g is gain coefficient and α are internal losses. Thelongitudinal propagation constant βz is

βz = k0 · ng (9.5)

where k0 = 2πλ

and ng effective group index

ng = c

vg(9.6)

Here vg is the group velocity. Applying the standing wave condition to an electromagneticwave of wavelength λ in a resonator of length L gives

L = mλ

2(9.7)

Expressions for transmitted components are

E1 = E0e− jβzL√

Gs

√1 − R1

√1 − R2

E2 = E0e− jβz·3L(√

Gs

)3√1 − R1

√R2

√R1

√1 − R2

E3 = E0e− jβz·5L(√

Gs

)5√1 − R1R2R1

√1 − R2

Total field, which is determined as a sum of all the above components, is therefore

Eout = E0

√(1 − R1) (1 − R2) Gse

− jβzL{1 + Gs

√R1R2e− jβz·2L

+ G2s R1R2e− jβz·4L + · · · }

Summing geometrical series, one finally obtains

Eout =√

1 − R1√

1 − R2GsE0e− jβzL

1 − Gs√

R1R2e−2 jβz·L


frequency

gain

Gmax

0νν

Fig. 9.4 Gain spectra showing frequency corresponding to gain peak.

Finally, gain of SOA with facet reflectivities R1 and R2 is

G = (1 − R1) (1 − R2) GS(1 − √

R1R2GS

)2 + 4√

R1R2GS sin2 φ(9.8)

Phase shift φ is obtained by assuming that the phase of the incident wave is taken as zero.The relation for phase change φ at the output is thus

φ = 2π

λL (9.9)

Using (9.6), we can write the above relation for frequency as

ν = φvg

2πL(9.10)

The above relation can be interpreted on gain spectrum graph as shown in Fig. 9.4 wherefrequency ν0 corresponds to gain peak Gmax, which is obtained when sin φ = 0, or φ = kπ .Using an expression for phase (9.9), one finds its value corresponding to ν0 (using relation1/λ0 = ν0/vg)

φ = 2π

λ0L = 2π

vgν0 L

or

ν0 = vg

2L· k

= vg

2L

assuming k = 1. The difference in frequencies ν − ν0 is determined as

ν − ν0 = vg

2πLφ − vg

2L= vg

2L

(φ

π− 1

)or

2π (ν − ν0 ) L

vg= π

(φ

π− 1

)= φ − π

Finally

φ = 2π (ν − ν0 ) L

vg(9.11)

where we have neglected π since sin π = 0.

227 General discussion

wavelength

gain

Fig. 9.5 Typical gain spectra of Fabry-Perot amplifier showing gain ripples.

10−4 10−3 10−210 0

101

102

103

Gai

n rip

ple

Reflectivity (R)

= 200Gs Gs = 500

Fig. 9.6 Gain ripple as a function of reflectivity for two values of gain.

9.1.2 The effect of facet reflectivities

An uncoated SOA has facet reflectivities (due to FP reflections) determined by taking thevalues of refractive indices of a typical semiconductor and the air and is approximatelyequal to 0.32. Even with the AR coatings, there are some residual reflectivities which resultin the appearance of the so-called gain ripples, see Fig. 9.5. Ripples are superimposed ongain spectrum. The peak-to-valley ratio between the resonant and non-resonant gains isknown as the amplifier gain ripple Gr. From Eq. (9.8) one obtains

Gr = 1 + Gs√

R1R2

1 − Gs√

R1R2(9.12)

where Gs is the single-pass amplifier gain.For an ideal TWA both R1 and R2 are zero. In this case Gr = 1; i.e. no ripple occurs at the

cavity mode frequencies. The quantity Gr is plotted in Fig. 9.6 as a function of reflectivity


R (assuming that R1 = R2) for two values of gain. One observes that gain ripple increaseswith increasing gain and also increasing facet reflectivity. The MATLAB code is providedin Appendix, Listing 9A.1.

9.2 SOA rate equations for pulse propagation

In this section we concentrate on basic rate equations describing pulse propagation inSOA [2], [4]. Discussion involves electromagnetic field and carriers. We start with thedevelopment of electromagnetic equation.

Pulse propagation inside SOA is described by the wave equation

∇2E (r, t ) − ε

c2

∂2E (r, t )

∂t2= 0 (9.13)

where E (r, t ) is the electric field vector, c is the light velocity and ε is the dielectric constantof the amplifier medium which is expressed as

ε = n2b + χ (9.14)

Here nb is the background refractive index of the semiconductor and susceptibility χ whichrepresents the effect of charges inside an active region in the phenomenological model is

χ = − cn

ω0(αH + i) g(n) (9.15)

where n is the effective mode index, αH is the linewidth enhancement factor (Henry factor)and g(n) is the optical gain approximated here (for bulk devices) as

g(n) = a (n − n0) (9.16)

where a is defferential gain, n is the injected carrier density and n0 is the carrier density attransparency.

In the following we assume a travelling wave semiconductor wave amplifier whichsupports single mode propagation with a perpendicular electric profile described by F (x, y).The electric field E (r, t ) which obeys wave equation (9.13) is expressed as

E (r, t ) = n1

2

{F (x, y) A(z, t) ei(k0z−ω0t)

}(9.17)

where n is the polarization vector, k0 = nω0/c and A(z, t) is the slowly varying amplitudeof the propagating wave. We introduce the following notation:

∇2 = ∇2⊥ + ∂2

∂z2, where ∇2

⊥ = ∂2

∂x2+ ∂2

∂y2

and evaluate derivatives

∇2E ∼ (∇2⊥F )Aeik0z + F

∂2A

∂z2eik0z + 2F

∂A

∂zik0eik0z + FA(ik0)

2eik0z (9.18)

229 SOA rate equations for pulse propagation

In the slowly varying envelope approximation (SVEA) the term with second derivative withrespect to z and t is neglected. Substituting (9.18) into (9.13), applying SVEA with respectto z and t and integrating over the transverse direction gives

∇2⊥F + ω2

0

c2

(n2

b − n2)

F = 0 (9.19)

and

∂A

∂z+ 1

vg

∂A

∂t= iω0�

2cnχA − 1

2αlossA (9.20)

where we have accounted for losses described by αloss. The group velocity is defined asvg = c/ng and group index ng is

ng = n + ω0∂n

∂ω(9.21)

The confinement factor � is

� =∫ w

0 dx∫ d

0 dy|F (x, y)|2∫ +∞−∞ dx

∫ +∞−∞ dy|F (x, y)|2 (9.22)

where w and d are the width and thickness of the amplifier active region. At this stage, onesimplifies Eq. (9.20) by introducing transformation to a reference frame moving with pulseas

τ = t − z

vg

z′ = z

In the new reference frame Eq. (9.20) takes the form

∂A

∂z= iω0�

2cnχA − 1

2αlossA (9.23)

Rate equation for carrier density n is in the following form [2], [4]:

dn

dt= I

qV− n

τc− �g(n)

�ω0σm|A|2 (9.24)

where I is the injection current, V is the active volume, q is the electron charge, τc is thecarrier lifetime and σm is the cross section of the active region. In the new reference frame,the above equation is

dn

dτ= I

qV− n

τc− �g(n)

�ω0σm|A|2 (9.25)

Slowly varying amplitude A(z, t) of the propagating wave is expressed as

A =√

Peiφ (9.26)


where P(z, τ ) and φ(z, τ ) are the instantaneous power and the phase of the propagatingpulse. Using the above equations, one obtains [2]

∂A

∂z= 1

2(1 + iαH ) g · A (9.27)

dg

dτ= −g − g0

τc− gP

Esat(9.28)

∂P

∂z= (g − αloss) P (9.29)

∂φ

∂z= −1

2αH g (9.30)

The quantity Esat is defined as Esat = τcPs, where Ps is the saturation power of the amplifier

Ps = hω0σm

a�τc(9.31)

In the above g0 is the small signal gain

g0 = �a

(Iτc

eV− n0

)(9.32)

Finally, the cross section σm of the active region is σm = wd.

Pulse amplification

Using previously defined equations, we will now analyse pulse propagation assuming zerolosses, i.e. αloss = 0, and also assuming that τp � τc, where τp is the width of the input pulse.Under this approximation pulse is so short that gain has no time to recover. Observing thatτc = 0.2–0.3 ns for typical SOA [4], this approximation works for τp equal to about 50 ps.Under the above approximations, one can obtain the analytical solution of the amplifierequations. One first integrates Eq. (9.29) to obtain output power as

Pout (τ ) = Pin(τ )eh(τ ) ≡ Pin(τ )G(τ ) (9.33)

where Pin(τ ) is the input power, and

h(τ ) =∫ L

0g(z, τ )dz (9.34)

Quantity h(τ ) is known as the total integrated net gain. Replacing the last term in Eq. (9.28)using (9.29) gives

dg

dτ= g0 − g

τc− 1

Esat

dP

dz

Integrating the above over amplifier length and using (9.33) gives

dh(τ )

dτ= g0L − h(τ )

τc− 1

EsatPin(τ ) [G(τ ) − 1] (9.35)

231 Design of SOA

The solution of the previous equation is [2], [4]

G(τ ) = eh(τ ) = G0

G0 − (G0 − 1) exp [−E0(τ )/Esat ](9.36)

where G0 is the unsaturated gain of the amplifier and the quantity E0(τ ) is given by

E0(τ ) =∫ τ

−∞Pin(τ

′)τ ′ (9.37)

E0(τ ) represents the fraction of the pulse energy contained in the leading part of the pulseup to τ ′ ≤ τ .

The above solution shows that due to time dependence of the gain, different parts ofinput pulse experience different amplification which leads to a modification of pulse shapeafter being amplified by SOA.

In the following we will restrict our analysis to the Gaussian input pulse

Pin(τ ) = P0 exp(−τ 2

)(9.38)

For the Gaussian input pulse, the expression for the quantity E0(τ ) can be found in a closedform as

E0(τ ) = P0τ01

2

√π [1 + er f (τ )] (9.39)

where er f (τ ) is the error function defined as

er f (τ ) = 2√π

∫ τ

0e−x2

dx (9.40)

We have also used the following property of error function [5]:

1 − er f (τ ) = 2√π

∫ ∞

τ

e−x2dx (9.41)

In Fig. 9.7 we plotted pulse shape for several values of unsaturated gain G0. One canobserve that the amplified pulse becomes asymmetric; i.e. its leading edge is sharpercompared to its trailing edge.

9.3 Design of SOA

Here, we briefly describe the main effects which must be considered when designing SOAwith proper characteristics. Out of many issues related to the design of SOA, we brieflyconcentrate here on two: suppression of cavity resonances and polarization insensitivity.Consult the literature [1], [2] and [6] for more information.

Suppression of cavity resonance

To fabricate a travelling wave SOA, the Fabry-Perot cavity resonances must be suppressed.To accomplish this, the reflectivities at both facets must be reduced. Three approaches were


−3 −2 −1 0 1 2 30

0.2

0.4

0.6

0.8

1

τ

Nor

mal

ized

out

put p

ower

Fig. 9.7 Output pulse shapes for several values of the unsaturated gainG0 = 10, 100, 1000 (increasing values to theleft).

used to achieve this goal (see Fig. 9.8): (a) to put anti-reflection (AR) coating at both facets,(b) to tilt the active region and (c) to use transparent window regions.

The principles of AR coating were discussed in Chapter 3. AR coating only works for aparticular wavelength and it is not suitable for a wide bandwidth. The analysis [7] showsthat with appropriate combination of the previously discussed methods it is possible toobtain an effective facet reflectivity of less than 10−4. Multilayer coatings can broadenwavelength range where there is low reflectivity.

Polarization insensitive structures

The state of polarization of an electric field during propagation in optical fibre changesrandomly. Therefore, after propagation when signal is ready for amplification (say, inSOA), its state of polarization is unknown. For amplifications of such signals it is thereforedesirable that SOA has polarization independent amplification. The main factor responsiblefor polarization sensitivity is the difference between confinement factors for TE and TMmodes [1]. Proper design of polarization insensitive SOA involves several techniques whichare summarized in [1].

Additionally, in modern SOAs which are based on quantum well (QW) designs instead ofbulk structures, due to their significantly reduced threshold current and increased efficiencythere exist additional polarization related effects associated with QW. QW significantlysuffers from polarization sensitivity, that is, a significant difference in gain between thetransverse-electric (TE) and transverse-magnetic (TM) polarization modes. This is a majorconcern as the polarization of a signal cannot be always controlled. Several approacheswere discussed to the design of polarization insensitive SOA [8], [9], [10]. The issue is ofsignificant importance for SOA built using quantum wells.

233 Some applications of SOA

Balksemiconductor

Opticalinputsignal

(a)

Amplified

Active region

AR coatingAR coating

opticalsignal

Opticalinputsignal

Opticalinputsignal

transparentwindow

transparentwindow

(b)



(c)

Fig. 9.8 Main ways to make travelling wave SOA.

9.4 Some applications of SOA

9.4.1 Wavelength conversion

One of the important applications of SOA is for wavelength conversion (WC) [11], seealso [1]. Three main methods of WC have been analysed: cross-gain modulation (XGM),cross-phase modulation (XPM) and four-wave mixing (FWM). XGM will be discussed indetail in the following section.

Other methods (not discussed here) include those based on the nonlinear optical loopmirror (NOLM) with the nonlinearity achieved by using fibre or SOA (see recent review[12]).


SOAλCW

λCWλSsignal

laser probe

output

Fig. 9.9 Wavelength conversion using XGM in SOA in copropagation configuration.

SOAλSsignal

output λCW

λCW

laser probe

Fig. 9.10 Wavelength conversion using XGM in SOA in a counter-propagation configuration.

Gain (dB)

Input signal power (dBm)

Converted signal, λCW In

put s

igna

l, λ S

Fig. 9.11 Principle of wavelength conversion in SOA using XGM.

Cross-gain modulation

The reduction of gain, known as gain saturation, typically occurs for input powers ofthe order of 100 µW or higher. To understand this effect, one must remember that theamplification in SOA is the result of stimulated emission. The rate of stimulated emissionin turn depends on the optical input power. At high optical injection, the carrier concentrationin the active region is depleted through stimulated emission to such an extent that the gainof SOA is reduced.

WC based on XGM can operate in either copropagation or counter-propagation config-urations. Counter-propagation configuration does not require optical filtering of the targetwavelength. However, counter-propagation suffers from speed limitations [13].

The schematic representation of the XGM used as wavelength conversion is shown inFig. 9.9 (copropagation configuration) and in Fig. 9.10 (counter-propagation configuration).The principle of wavelength conversion employing nonlinear characteristics of SOA isexplained in Fig. 9.11. Two optical signals enter a single SOA. One of the signals (knownas the probe beam) at wavelength λCW is injected continuously (CW); the other (known asthe signal beam) injected at wavelength λS is carrying digital information.

235 Some applications of SOA

A

B

Δφ

SOA 2

SOA 1C

A1

A2

Fig. 9.12 The MZ interferometric configuration and its equivalent symbol.

If the peak optical power in the modulated signal is near the saturation power of the SOA,the gain will be modulated synchronously with the power. When the data signal is at highlevel (a logical ONE), the gain is depleted and vice versa. This gain modulation is imposedon the unmodulated (CW) input probe beam. Thus, an inverted replica of the input data iscreated at the probe beam wavelength. This form of wavelength conversion is one of thesimplest all-optical wavelength conversion mechanisms available today.

Very fast wavelength conversion can be achieved with speeds in the order of 40 Gb s−1

[14] with a small bit error ratio penalty. Previously, it was thought that the speed of WC islimited by the intrinsic carrier lifetime, which is around 0.5 ns. This is because the effectivecarrier lifetime can be decreased by the use of high optical injection to values as low as10 ps.

Another effect associated with pulse propagation in long SOA (>1 mm) is that pulsedistortion of the input data due to gain saturation effects can tend to sharpen the leadingedge of the data pulse at the probe wavelength.

9.4.2 All-optical logic based on interferometric principles

For the future high-speed optical networks, several all-optical signal processing functional-ities will be required to avoid cumbersome and power consuming electro-optic conversion.The central role in this process is played by all-optical high-speed logic gates.

In recent years several schemes of implementation have been investigated. Some of thoseschemes exploit gain saturation of SOA, and the other methods employ the interferometricconfigurations.

Optical logic devices are constructed using the Mach-Zehnder (MZI) interferometer, seeFig. 9.12 for an introduction to the notation and symbols. �φ is the phase difference betweensignals propagating in the upper and lower arms of MZI. Each arm of the interferometercontains SOA where the effect of cross-phase modulation (XPM) is utilized to change phaseof the transmitted light. The XPM effect is based on the physical effect of the dependenceof refractive index on the carrier density in the active region. Depending on operatingconditions which are controlled by driving currents within each SOA and direction andintensity of external light, this configuration can perform like an all-optical gain. In whatfollows, for several logic gates we schematically show operating conditions of MZI, theresulting phase difference and the equivalent logical table [15].


A

B

XXBA

00

00

000

11

1

11

Δφ = 0

Fig. 9.13 A block diagram of A NOT B and a truth table.

A

B

XXBA

0 0

00

000

11

1

11

Δφ = π

Fig. 9.14 A block diagram of A AND B and a truth table.

A and NOT B

Here, bias conditions are set in such way that, when a second signal B is input in a counter-propagating scheme, the phase difference is zero. The configuration and logical table areshown in Fig. 9.13.

A AND B

Here, both SOA are biased such that in the absence of counterpropagating signal B, a phasedifference of π exists at the output X . The relevant configuration and the truth table areshown in Fig. 9.14.

9.5 Problem

1. Derive an expression for gain ripple Gr.

9.6 Project

1. Use software developed by Connelly [1] to analyse optical logical elements.



237 Appendix 9A



9A.1 gain ripple.m Determines gain ripples as a function of reflectivity9A.1 pulse shape.m Determines output pulse shape for various values of G0

Listing 9A.1 Program gain ripple.m. MATLAB program which determines gain ripplesfor SOA.

% File name: gain_ripple.m

% Calculates gain ripple as a function of reflectivity

clear all


R = linspace(1d-4,4d-3,N_max); % creation of theta arguments

%

for G_s = [200 500] % gain coefficient

G_r = (1 + G_s.*R)./(1 - G_s.*R);

loglog(R,G_r,’LineWidth’,1.5)

hold on

end


ylabel(’Gain ripple’,’FontSize’,14)

xlabel(’Reflectivity (R)’,’FontSize’,14)


text(3d-3, 12, ’G_s = 200’,’Fontsize’,14)

text(6d-4, 12, ’G_s = 500’,’Fontsize’,14)

text(8.5, 0.1, ’r = 0.8’,’Fontsize’,14)

pause

close all

Listing 9A.2 Program pulse shape.m. MATLAB program which determines output pulseshape for three values of the unsaturated gain G0.

% File name: pulse_shape.m

% Determines output pulse shapes in soa

% Basic equations after Agrawal and Olsson, JQE25, 2297 (1989)

clear all

P_0 = 1;

w = 0.1; % defined as P_0*tau_0/E_sat

tau = -3:0.01:3;

hold on

for G_0 = [10 100 1000]

x = (1/2)*sqrt(pi)*w*(1 + erf(tau));


G = G_0./(G_0 -(G_0-1)*exp(-x));

y = exp(-tau.^2).*G;

y_max = max(y);

plot(tau,y/y_max,’LineWidth’,1.5)

end

xlabel(’\tau’,’FontSize’,14);

ylabel(’Normalized output power’,’FontSize’,14);


grid on

pause

close all

References

[1] M. J. Connelly. Semiconductor Optical Amplifiers. Kluwer Academic Publishers,Boston, 2002.

[2] N. K. Dutta and Q. Wang. Semiconductor Optical Amplifier. World Scientific,Singapore, 2006.

[3] M. J. O’Mahony. Semiconductor laser optical amplifiers for use in future fiber systems.J. Lightwave Technol., 6:531–44, 1988.

[4] G. P. Agrawal and N. A. Olsson. Self-phase modulation and spectral broadeningof optical pulses in semiconductor laser amplifiers. IEEE J. Quantum Electron.,25:2297–306, 1989.

[5] M. R. Spiegel. Mathematical Handbook of Formulas and Tables. McGraw-Hill, NewYork, 1968.

[6] G. P. Agrawal and N. K. Dutta. Semiconductor Lasers. Second Edition. KluwerAcademic Publishers, Boston/Dordrecht/London, 2000.

[7] T. Saitoh, T. Mukai, and O. Mikame. Theoretical analysis and fabrication of antire-flection coatings on laser-diode facets. J. Lightwave Technol., 3:288–93, 1985.

[8] Y. Zhang and P. P. Ruden. 1.3−µm polarization-insensitive optical amplifier structurebased on coupled quantum wells. IEEE J. Quantum Electron., 35:1509–14, 1999.

[9] W. P. Wong and K. S. Chiang. Design of waveguide structures for polarization-insensitive optical amplification. IEEE J. Quantum Electron., 36:1243–50, 2000.

[10] M. S. Wartak and P. Weetman. The effect of thickness of delta-strained layers in thedesign of polarization-insensitive semiconductor optical amplifiers. IEEE Photon.Technol. Lett., 16:996–8, 2004.

[11] T. Durhuus, B. Mikkelsen, C. Joergensen, S. L. Danielsen, and K. E. Stubkjaer.All-optical wavelength conversion by semiconductor optical ampliers. J. LightwaveTechnol., 14:942–54, 1996.

[12] I. Glesk, B. C. Wang, L. Xu, V. Baby, and P. R. Prucnal. Ultra-fast all-optical switchingin optical networks. In E. Wolf, ed., Progress in Optics, volume 45, pp. 53–117.Elsevier, Amsterdam, 2003.

239 References

[13] D. Nesset, T. Kelly, and D. Marcenac. All-optical wavelength converion using SOAnonlinearities. IEEE Communications Magazine, December:56–61, 1998.

[14] D. Wolfson, A. Kloch, T. Fjelde, C. Janz, B. Dagens, and M. Renaud. 40-gb/sall-optical wavelength conversion, regeneration, and demultiplexing in and SOA-based all-active Mach-Zehnder interferometer. IEEE Photon. Technol. Lett., 12:332–4, 2000.

[15] K. Roberts-Byron, J. E. A. Whiteaway, and M. Tait. Optical logic devices and methods.United States Patent, Number 5,999,283. Dec. 7, 1999.

10 Optical receivers

In this chapter we summarize the operation of an optical receiver, which is an importantpart of an optical communication system. An overview of design principles for receiversused in optical communication systems is provided by Alexander [1]. The aim of a receiveris the recovery of the transmitted data. The process involves two steps [2]:

1. the recovery of the bit clock,2. the recovery of the transmitted bit within each bit interval.

A block diagram of an optical receiver is shown in Fig. 10.1 [3], [4].The receiver consists of a photodetector which converts the optical signal into electrical

current. A good light detector should generate a large photocurrent at a given incident lightpower. They should also respond fast to the input changes and add minimal noise to theoutput signal. This last requirement is of crucial importance since the received signal istypically very weak. In digital optical communication systems the detection process is oftenconducted with a PIN photodiode.

There are generally two types of detection [4]: direct detection (also called incoherentdetection) and coherent detection.

Direct detection detects only the intensity of the incident light. It is used mainly forintensity or amplitude modulation schemes. It can only detect an amplitude modulated(AM) signal.

Coherent detection can detect both the power and phase of the incident light. It is thereforeused when phase modulation (PM) or frequency modulation (FM) is preferred. Coherentdetection is also important in applications such as WDM.

The coherent detection requires a local oscillator to coherently down-convert the modu-lated signal from optical frequency to intermediate frequency (IF) [4]. Coherent detectionis not discussed here, see [4] for the extensive discussion. The incoherent detection whichdominates in currently deployed systems is based on square-law envelope detection of theoptical signals.

After detection, the electrical current is often amplified (not shown in Fig. 10.1) andthen passes through an electrical filter which is normally of the Bessel type. At that point,electrical eye diagrams are typically observed for the assessment of signal quality. Next,sampling of electrically filtered received signal is performed. The received electrical signalis corrupted with noise of various origins. The noise sources will be discussed in thefollowing sections.

Performance evaluation of an optical transmission system is done by evaluating opticalsignal-to-noise ratio, eye opening and bit error rate (BER) which is the ultimate indicatorof the system’s performance.

240

241 Main characteristics

Photodetector

Inputsignal

OutputsignalReceiver filter

hR (t)Sampler/detector

i(t)

Fig. 10.1 Block diagram of an ideal optical receiver.

10.1 Main characteristics

Before we enter a more specialized discussion of optical receivers, let us summarize theirmain characteristics [5], which are: receiver sensitivity, dynamic range, bit-rate transparencyand bit-pattern independency.

10.1.1 Receiver sensitivity

This property is a measure of the minimum level of optical power Psens at the receiverrequired for a reliable operation. Specifically, one expects that the BER is smaller than aspecified level. Typically, that level is established to be equal to 10−9.

The receiver sensitivity is a function of both the signal and also the noise parametersof photodetector and a preamplifier. It is a measure of the operating limit of the opticalreceiver, which, however, rarely operates close to that limit. There always exists a possibilityof degradation of the system (temperature, ageing, etc.), so a typical margin (normally 3–6 dB) is established.

Receiver sensitivity is a fundamental parameter of an optical receiver. It is directlyresponsible for the spacing between two points in any optical link, e.g. between transmitterand receiver or between repeaters.

10.1.2 Dynamic range

Dynamic range (expressed in dB) is the difference between the maximum allowable powerand minimum power determined by receiver sensitivity. The maximum allowable power onthe receiver is determined by nonlinearity and saturation.

Large dynamic range is important because it allows for more flexibility in the design ofan optical network. The design of every network should take into account the wide rangeof possible changes of received optical powers due to changes in temperature, ageing orvarious types of losses (in the fibre, connectors, etc.).

10.1.3 Bit-rate transparency

It refers to the ability of the optical receiver to operate over a range of bit rates. It describesthe ability of the same receiver to be used for several networks operating at different bitrates.


10.1.4 Bit-pattern independency

This is the property of an optical receiver determining its operation for various data formats.The main constraint is imposed by non-return-to-zero (NRZ) code.

10.2 Photodetectors

At the end of travel through a optical fibre, the optical signal reaches a photodetector (PD)where it is converted into an electrical signal. In PD photon of energy hν is absorbed andproduces photocurrent iP:

iP = ηP

hνe (10.1)

where P is the power of the incoming light, η is the quantum efficiency, h is Planck’s constantand ν = c/λ is the frequency of the absorbed light. PD is similar to a semiconductor diodepolarized in the reverse direction. Therefore, PD can be modelled as a current source.

Another very popular detector is a human eye. It is the most popular natural detector oflight but it has several disadvantages: it is slow, has bad sensitivity for low-level signals,has no natural connection to electronic amplifiers and its spectral response is limited to the0.4–0.7 µm range.

Artificial (human made) optical detectors are based on two physical mechanisms: externalphotoelectric effect and internal photoelectric effect. In the external photoelectric effect,electrons are removed from the metal surface of electrode known as a cathode by absorbingenergy from incident light. Then, under an electric field due to the potential differencebetween both electrodes, they travel to another electrode known as an anode, thus producingelectrical current. Vacuum photodiodes and photomultiplier tubes operate on that principle.

Main choices for photodetectors are the PIN (P-type Intrinsic N-type) photodiode andavalanche photodiode (APD). APD provides gain which increases system sensitivity butintroduces more noise.

In this section we will discuss only the internal photoelectric effect and its applications inlight detection. In this effect, physical processes take place inside semiconductor junctiondevices. There, free carriers (electrons and holes) are generated by absorption of incomingphotons, and as a result an electrical current is produced. These devices can be viewed asthe inverse of a light emitting diode (LED).

10.2.1 Principles of photo detection

Photodetection using semiconductors is possible because of optical absorption. When lightis incident on the semiconductor surface, it may or may not be absorbed depending on awavelength. Absorbed optical power is described as

dP

dx= α(λ)P (10.2)

243 Photodetectors

Abs

orpt

ion

coef

ficie

nt (c

m-1

)

Wavelength λ (μm)

0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.810 1

10 2

10 3

10 4

10 5

Si

Ge

GaAs

Fig. 10.2 Optical absorption coefficient as a function of wavelength for typical semiconductors. Reprinted from H. Melchior,Journal of Luminescence, 7, 390 (1973). Copyright (1973), with permission from Elsevier.

where α(λ) is the absorption coefficient which is wavelength dependent. The spectra ofoptical absorption for several semiconductors (including compounds) are schematicallyshown in Fig. 10.2. As can be seen, the absorption coefficient α(λ) strongly depends onwavelength.

Integration of Eq. (10.2) gives the absorbed power at the distance x from the surface interms of the incident optical power P0:

P(x) = P0(1 − e−α(λ)·x) (10.3)

A schematic of a photodetector is illustrated in Fig. 10.3. Incident light penetrates thesemiconductor surface and generates electrical current I . The device structure consists ofp and n regions separated by wide intrinsic (very lightly doped) region of width w underlarge reverse bias voltage, see Fig. 10.4.

When a photon is absorbed by the solid (and if it has enough energy, i.e. large enoughfrequency), its energy causes an electron to move from valence band (leaving one holebehind) to the conduction band and to create one extra electron there. The change of energyof an electron should be (at least) Eg, which is known as the energy gap. Photons withsmaller energies are usually not able to create electron-hole pairs, see Fig. 10.5. Therefore,photon energy Ep = hν must be

Ep ≥ Eg or hν ≥ Eg or λc = ch

Eg


Incident light

I

Electrode

Semiconductor

Fig. 10.3 Schematic of a photodetector.

photon hν

electronholep n

R

i

V

Fig. 10.4 PIN photodetector in external electrical circuit.

+

p-type n-type

Ev

Ec

Depletionregion

Eg

Photonhν ≥ Eg

Intrinsicregion

Fig. 10.5 Energy band diagram of a PIN photodiode. Generation of electron-hole pair is shown.

245 Photodetectors

Table 10.1 Bandgaps and cutoff wavelengths for different semiconductors and their compounds.

Semiconductor �E(eV ) λc(µm)

C 7 0.18Si 1.1 1.13Ge 0.72 1.72Sn 0.08 15.5GaxIn1−xAs 1.43–0.36 0.87–3.44GaxIn1−xAsyP1−y 1.36–0.36 0.92–3.44

Rj

ip

RsVLRLCj

Fig. 10.6 Equivalent circuit model of a PIN photodetector.

where λc is known as cutoff and it defines a usable spectrum. The practical formula is

λc = 1.24

Eg, λc is in µm and Eg in eV

The applied high electric field causes electrons and holes to separate. They subsequentlyflow to n (electrons) and p (holes) regions and produce flow of current in the external circuitknown as photocurrent.

Different semiconductor materials have different bandgaps and therefore different cutoffwavelengths. They are used as detectors in different parts of the electromagnetic spectrum.Bandgaps and cutoff wavelengths for some semiconductors and compounds are summarizedin Table 10.1.

An equivalent circuit of PD is shown in Fig. 10.6 [1]. It consists of a current source,parallel junction resistance Rj , which is equivalent to the differential resistance of the diode.Rj is typically large, around 106 �, and usually it is ignored in the analysis. Cj is the junctioncapacitance; Rs is the series resistance due to bulk and contact resistance and typically is afew ohms. RL is the load resistance. In a typical analysis Rj and Rs are neglected.

The spectral function VL(ν) of the output voltage is related to the spectral function of thephotocurrent iP(ν) as

VL(ν) = iP(ν)RL

1 + j2πνRLCj(10.4)


(a) (b)

Si-InP substrate

InP buffer

Electricfield

Schottkyjunction

Incident light

Graded layer

InGaAs(Photon absorption)

Ti/Au metal contactsInAlAs

+ –

Fig. 10.7 Schematic of a metal-semiconductor-metal (MSM) photodetector. Top view of the structure (a) and its cross-sectionview (b).

The previous equation is obtained by solving Kirchoff’s loop equation. Equivalentimpedance is

1

Ze= 1

RL+ 1

1jωCj

= 1

RL+ jωCj = 1 + jωCjRL

RL

Using VL(ν) = iP(ν) ·Ze, Eq. (10.4) follows. Frequency bandwidth is therefore determinedby the time constant CjRL. One wants to keep this time constant small in order to achievea large bandwidth.

We finish this section with a brief discussion of an integrated optical receiver usinga metal-semiconductor-metal (MSM) photodetector, which is shown in Fig. 10.7. TheMSM detector structure consists of an interdigital pattern of metal fingers deposited on asemiconductor substrate and a typical vertical PIN detector [6]. Electric potential is appliedbetween alternate fingers creating an electric field which sweeps photogenerated electronsand holes to the positive and negative electrodes, respectively.

This structure shows several improvements compared to traditional designs, like a signif-icant improvement in sensitivity. Most of the improvements result from the lateral design.

10.2.2 Performance parameters of photodetectors

Main parameters which determine characteristics of PD are [7]: dark current, spectralresponse, quantum efficiency, noise, detectivity, linearity and dynamic range and speed andfrequency response. We will now discuss them in more detail.

Dark current

Dark current is the current flowing in the PD without incoming light. Typical values forpopular semiconductors are shown in Table 10.2.

247 Photodetectors

Table 10.2 Dark currents for different semiconductors and their compounds.

Semiconductor Dark current (nA)

Si 0.1–1Ge 100InGaAsP 1–10

Quantum efficiency

Quantum efficiency is a measure of the efficiency of the generation of the electron-holepairs (EHP). It is defined as [8]

η = number of free EHP generated

number of incident photons= Iph/e

P0/hν(10.5)

Here Iph is the photocurrent flowing in the external circuit, e is the electron’s charge, P0 isthe incident optical power and hν is the energy of a single photon.

Responsivity

It is also known as spectral responsivity. It is defined as [8]

R = Iph

P0(10.6)

Responsivity specifies the photocurrent generated per unit optical power. It can be expressedas

R = ηe

hν= η

eλ

hc(10.7)

Ideal responsivity is therefore a linear function of wavelength λ, see Fig. 10.8.

Speed of response

It determines how PD responds to an optical signal. A typical response to a pulse is shownin Fig. 10.9.

Speed of response is determined by the RC time constant. In terms of parameters ofpreviously introduced equivalent circuit, the rise time τr is given by

τr = 2.19 · RL · Cj

The evaluation of τr is left as an exercise. Time response is directly related to the frequencyresponse. The 3-dB bandwidth is [9]

fs−dB = 1

2πRLCj


Wavelength (μm)

Res

pons

ivit

y (A

/W)

Ideal photodetector= 0.9

1.10.7 0.9

0.2

0

0.4

0.8

1.0

0.6

Si

η

Fig. 10.8 Typical current responsivity versus wavelength for Si. Also shown is the responsivity of an ideal photodiode withquantum efficiency η = 0.9.

t

190%

t

τ

is

Ps

r

Fig. 10.9 Typical response of a photodetector to a square-pulse signal.

10.2.3 Photodetector noise

Typical optical signals arriving at the receiver front end are very weak. They are, therefore,significantly affected by various noise sources.

There are several physical processes which contribute to noise. For example, the APDgenerates noise in the avalanche process and optical amplifiers produce noise due to am-plified spontaneous emission (ASE). In addition, there is always a quantum noise whichexists in all devices and sets a fundamental lower limit on noise power.

The working parameter characterizing the photodetector is the signal-to-noise ratio(SNR), which is defined as [10]

S

N= signal power from photocurrent

photodetector noise power + amplifier noise power(10.8)

249 Photodetectors

Noise appears as random fluctuations in a signal. A typical measure of noise is associatedwith the variance or the mean square deviation of the signal s, which is defined as [7]

σ 2s = (s − s)2 = s2 − s2

The mean value (average) s is defined as

s =∑

s

p(s) s

where p(s) is the probability of the measured signal having a value s and the sum isevaluated over all possible values obtained from measuring the signal.

Noise in a signal s can be represented by random variable sn:

sn = s − s

If in the device (or system) there are two or more simultaneously present noise sourcessn1, sn2, . . . which are independent, their combined effect is found by adding their meansquare values (or their powers):

s2n = s2

n1 + s2n2 + · · ·

In terms of the above quantities, the SNR is expressed as

SNR = s2

s2n

= s2

σ 2s

or SNR = 10 logs2

σ 2s

[dB]

Main types of noise arising in a photodetector will now be summarized.

Shot noise

Shot noise is due to random distribution of electrons generated in the photodetector. It isassociated with the quantum nature of photons arriving at the photodetector which gener-ates carriers. Photons arrive at the photodetector randomly in time due to their quantum-mechanical nature. Their randomness is described by Poisson statistics. The resultingexpression for shot noise of current in the photodetector is [4], [7]

i2s = 2eBis (10.9)

where e is the electron’s charge, B is the bandwidth and is is the average value of signalcurrent.

Thermal noise

Thermal noise or Johnson noise is due to the random motion of electrons in the resistor R.It is modelled as a Gaussian random process with zero mean and autocorrelation functiongiven by [4]

4kBT

Rδ(τ )


LR

is

Cj

Fig. 10.10 Equivalent circuit model of a PIN photodetector with an amplifier.

where δ(τ ) is the Dirac delta function. Thermal noise power in a bandwidth B is [7]

ps,th = 4kBT B (10.10)

where kB = 1.38×10−23 J/K is the Boltzmann’s constant and T is the absolute temperature.Thermal noise can be expressed as a current source [4], [7]

i2s,th = 4kBT

RB ≡ I2

T B (10.11)

Here IT is the parameter used to specify standard deviation in units of pA/√

Hz. Its typicalvalue is 1pA/

√Hz [2].

10.2.4 Detector design

In this section we perform a simple analysis of a PIN photodetector in the presence of noise[11]. Using results 10.6 and 10.7, the signal current of the detector is

is = ηeλ

hcP (10.12)

where P is the optical power coming to the detector at wavelength λ. Equivalent circuit ofa PIN photodetector with an amplifier is shown in Fig. 10.10. It is a simple modification ofthe circuit shown in Fig. 10.6.

Noise generated in the amplifier is essential for the operation. Amplifier noise is rep-resented [11] as Johnson noise originating in the load resistor RL. Using Eq. (10.11), thesignal-to-noise ratio is

S

N= i2s · RL

i2s,yh · RL

= i2s · RL

4kB · T · B(10.13)

where B is the bandwidth and T is the resistor’s temperature. The SN ratio cannot increase

indefinitely by increasing value of RL because of the effect of junction capacitance Cj . In

the typical situation, the bandwidth is determined by(RL · Cj

)−1. Using the expression for

the bandwidth B = (2π · RL · Cj

)−1(see problem), Eq. (10.13) takes the form

S

N= i2s

8kB · T · B2 · Cj(10.14)

251 Receiver analysis

106

107

108

109

1010

10−10

10−8

10−6

10−4

Detector bandwidth (Hz)

Min

imum

sig

nal p

ower

(W

)

500 K1500 K

Fig. 10.11 Plot ofPmin versus bandwidth for PIN detector for three values of temperature (1500 K, 1000 K and 500 K).

Minimum detectable signal is,min can be determined from the above by setting SN = 1. One

finds

is,min = 2 · B · (2π · kB · T · Cj

)1/2(10.15)

Minimum detectable optical power is determined from Eq. (10.12) as

Pmin = 2hc

ηeλ· B · (2π · kB · T · Cj

)1/2(10.16)

The plot of Pmin versus bandwidth B is shown in Fig. 10.11. The following parameters wereassumed: Cj = 1pF, η = 1, T = 1000 K, λ = 0.85 µm. MATLAB code is provided in anAppendix, Listing 10A.1.

10.3 Receiver analysis

Optical detection theory is extensively discussed, e.g. by Einarsson [12]. The problem in-volves detection of signals with noise. In optical systems the information is transmittedusing light which consists of photons. Due to their statistical nature, the transmitted in-formation will always show random fluctuations. Those fluctuations determine the lowestlimit on transmitted power. In addition, there exist other noise contributions which originatefrom various processes. Some of them have already been discussed.

Digital signal under consideration operates at a bit rate B. The time slot (or bit interval)T is

T = 1

B


and it is the inverse of the bit rate. The input data sequence in the communication systemis denoted by {bk}. The optical power p(t) falling on the photodetector is the sequence ofpulses and it is written as

p(t) =k=+∞∑k=−∞

bk hp(t − k · T ) (10.17)

where k is a parameter denoting the k-th time slot and hp(t) represents the pulse shape ofan isolated optical pulse at the photodetector input.

It is assumed that [5]

1

T

∫ +∞

−∞hp(t)dt = 1 (10.18)

so that bk represents the received optical power in the k-th time slot.Eq. (10.17) is based on the assumption that the system consisting of transmitter and

optical fibre is a linear one and time-invariant. bk in Eq. (10.17) can take two values b0

and b1, which correspond to logical values in the k-th time slot being ZERO or ONE. Inan ideal case one would expect the b0 to be zero so no optical power is transmitted for thelogical ZERO. However, semiconductor lasers always operate at the nonzero bias current,so there is always some small optical power transmitted.

10.3.1 BER of an ideal optical receiver

There are several ways to measure bit error rate (BER), that is the rate of error occurrencein a digital data stream. It is equal to number of errors occurring over some time intervaldivided by total number of pulses (both ones and zeros).

The simplest method to define BER is [13]

BER = Ne

Np= Ne

B · t(10.19)

where Ne is the number of errors appearing over time interval t, Np is the number of pulsestransmitted during that interval, B = 1/T is the bit rate and T is the bit interval.

The required BER for high-speed optical communication systems today is typically10−12, which means that on average one bit error is allowed for every terabit of datatransmitted. BER depends on the various signal-to-noise ratio (SNR) of the fibre system,like the receiver noise level.

Direct optical detection is a process of determining the presence or absence of lightduring a bit interval. No light is interpreted as logical ZERO; some of the light presentsignals is logical ONE.

In a real life, the detection process is not so simple because of the random nature ofphotons arriving at the receiver. Their arrival is modelled as a Poisson random process. Therandom process, in time arrivals of photons at the photodetector, is shown in Fig. 10.12.

For an ideal optical receiver we will assume that there are no noise sources in the system.The average number of photons arriving at the photodetector, with hνc being the energy of


time

T

Fig. 10.12 Random arrivals of photons at photodetector are described by the Poisson process. Each photon is represented by a boxand they all have the same amplitude.

a single photon, is thus

N = p(t)

hνc(10.20)

where p(t) is power of light signal, h is Planck’s constant and νc is the carrier frequency.The optical power impinging on the photodetector is expressed by Eq. (10.17).

A simple expression for BER for ideal receiver (no noise) can be obtained as follows(following Ramaswani and Sivarajan [2]).

The probability that n photons are received during a bit interval T is

e− NB

(NB

)n

n!

where N is the average number of photons given by Eq. (10.20). Probability of not receivingany photons (n = 0) is exp (−N/B). Assume equal probabilities of receiving ZERO andONE. The BER of an ideal receiver is thus

BER = 1

2e−N/B ≡ 1

2e−M (10.21)

where M = NB = p

hνcB represents the average number of photons received during one bit.Expression (10.21) represents BER for an ideal receiver and it is called the quantum

limit. To get a typical bit rate of 10−12, the average number of photons is M = 27 per onebit.

10.3.2 Error probability in the receiver

Assume that a binary signal current at the photodetector is [4]

itot(t) =+∞∑

n=−∞Bn · hp(t − nTB) + ID + inoise(t)

= iph(t) + ID + inoise(t)

where iph(t) is due to the amplitude modulated pulse signal, ID is the dark current of thephotodiode and inoise(t) is the noise current. Bn takes two values BH and BL correspondingto logical ONE and logical ZERO, respectively.

Assume (after Liu [4]) that H (ω) is the combined transfer function of the front-endreceiver and equalizer. The output at the equalizer is

yout (t) = itot(t) ⊗ h(t) (10.22)


where ⊗ denotes convolution and h(t) corresponds to the transfer function H (ω). Thesignal component at the output of the equalizer is

ys(t) = iph(t) ⊗ h(t) =+∞∑

n=−∞Bn · hp(t − nTB) ⊗ h(t)

≡+∞∑

n=−∞Bn · hp(t − nTB)

The signal output of the equalizer is sampled at the bit rate and compared with a threshold.This process allows for detection of the amplitudes Bn. The output signal after equalizerwithout (constant) dark current is

yout (t) =+∞∑

n=−∞Bn · pp(t − nTB)

where pp(t) contains signal and noise. The above is sampled and the output at timenTB + τ (0 < τ < TB) is

yout,n(t) ≡ yout (nTB + τ ) =∑

k

Bk · pp([n − k]TB + τ ) + ynoise,n

≡ Bn · pp[0] + ISIn + ynoise,n (10.23)

with the following definitions:

pp[n] = pp(nTB + τ )

ISIn =∑k �=n

Bk · pp([n − k])

and

ynoise,n = ynoise,out (nTB + τ )

is the sampled noise term. ISIn refers to the intersymbol interference.In a practical receiver, the decision as which bit (ZERO or ONE) was transmitted in each

bit interval is done by sampling current after equalizer. With the presence of noise, there isnonzero probability of an error. The errors are typically described by BER (bit error rate).

Introduce yth as a threshold used in error detection. From Eq. (10.23), one can observethat errors occur when

Bk = BH when yout,n < yth

or

Bk = BL when yout,n > yth

Therefore, the probability of an error detection is [14]

Perror = p0P(yout,n > yth|Bk = BL

)+ p1P(yout,n < yth|Bk = BH

)Here p0 and p1 are a priori probabilities for bits ZERO and ONE.


P(0|1)

P(1|0) VH

VL

Vthreshold

Fig. 10.13 Determination of bit error rate for a Gaussian process.

The decision taken involves comparison of the measured photocurrent I with a thresholdIth. If I > Ith, one decides that a ONE bit was transmitted; otherwise for I < Ith a bit ZEROwas transmitted.

10.3.3 BER and Gaussian noise

In optical digital communication the transmitted signal is never perfectly recovered. Noise,adjacent channel interference, amplified spontaneous emission, all result in creation oferrors. Additionally, pulses are broadened due to fibre dispersion, device nonlinearity orslow photodiode response. Created signal distortion is known as an intersymbol interference(ISI) and it arises when adjacent bits are corrupted. It results in possible detection errorsfor the transmitted bits. ISI is random because the binary bits in a digital signal are random.To minimize ISI, an equalizer is generally used.

We analyse the performance of real receiver with noise [15]. We want to establishconnection between the signal-to-noise ratio (SNR) and BER. By referring to Fig. 10.13,we assume that VH (high voltage at the output of the receiver) and VL (low voltage at theoutput of the receiver) are random variables described by the Gaussian probability densityfunction (pdf) as

p(x) = 1√2π

e− x2

2 (10.24)

BER is defined in terms of two conditional probabilities P(0|1) and P(1|0) as

BER = p0P(0|1) + p1P(1|0) (10.25)

where p0 and p1 are a priori probabilities for bits ZERO and ONE. Their values are p0 =p1 = 1

2 . Conditional probability P(A|B) for two events A and B is defined as

P(A|B) = P(A · B)

P(B)


3 4 5 6 7 810−20

10−15

10−10

10−5

100

Q

Bit

Erro

r Rat

e

Fig. 10.14 Plot of bit error rate versus Q.

The conditional probabilities P(0|1) and P(1|0) are shown in Fig. 10.13. Assuming Gaus-sian pdf, they are determined as

P(0|1) = 1√2π

∞∫Q1

e− x2

2 dx

P(1|0) = 1√2π

∞∫Q2

e− x2

2 dx

where functions Q1 = Vth−VL

σnoiseand Q1 = VH −Vth

σnoiseare integration limits. We assume that

distributions of ZERO and ONE are identical and that noise variance σnoise is the same forboth. It follows that the optimum threshold is 1

2 (VH + VL). Thus

Q = Q1 = Q2 = VH − VL

2σnoise

Using definition (10.25), the BER is evaluated as

BER = 1

2P(0|1) + 1

2P(1|0)

= 1√2π

∞∫Q

e− x2

2 dx

The above function relates BER to the signal-to-noise ratio Q. It is plotted in Fig. 10.14.For a typical BER of 10−6, Q = 6. MATLAB code used to create this figure is shown inAppendix, Listing 10A.2.

257 Modelling of a photoelectric receiver

The average incident input power P is defined as

P = 1

2(PH + PL) (10.26)

where PH and PL are optical powers corresponding to logical ONE and ZERO (high andlow), respectively. Those powers are related to voltages VH and VL introduced earlier as

VH = R · PH · ys,max(t) (10.27)

VL = R · PL · ys,max(t) (10.28)

where R is the responsivity of the detector and ys,max(t) is the maximum output voltage ofthe receiver during a given bit period. In terms of the above quantities, the average powerP is evaluated as follows:

P = 1

2(PH + PL) = 1

2

PH + PL

PH − PL(PH − PL)

= 1

2· 1 + PL

PH

1 − PLPH

· 1

R· 1

ys,max(t)· (VH − VL)

= 1

2· 1 + r

1 − r· 1

R· Q ·

√n2

out

ys,max(t)(10.29)

The above expression gives the average incident input power level necessary to obtain agiven BER. In the above derivation, we have used σ 2

noise = n2out , where n2

out is the varianceof the output noise. We also defined extinction ratio r as

r = PL

PH(10.30)

For a PIN photodiode, the receiver sensitivity takes the form

ηP = hν

q· 1 + r

1 − r· Q ·

√n2

out

ys,max(t)(10.31)

Ideal sensitivity is obtained where no power is transmitted (r = 0) as

ηP = hν

q· Q ·

√n2

out

ys,max(t)(10.32)

Using the above results, one can analyse receiver sensitivity for various types of noisefilters. We refer to the literature for a more detailed discussion [15], [14], [4].

10.4 Modelling of a photoelectric receiver

A schematic of the receiver is shown in Fig. 10.15. It is modelled by a filter function H (ν),see Geckeler [16]. As a front end it contains an avalanche photodiode, with M being themultiplication factor of the photocurrent and G the gain factor of an electronic amplifier.


PD MG

H(ν)S (ν)1 S (ν)2

s(t)P1

τ

Fig. 10.15 Block scheme of the receiver which is described by a filter functionH (ν)with parameter τ .

Both M and G are assumed to be frequency-independent. Frequency dependence of thereceiver is described by a low-pass filter function H (ν), which we assume to be a realfunction (not complex). Possible phase distortions which will result in function H (ν) beingcomplex can be compensated by an equalizer.

In Fig. 10.15 S1(ν) is the frequency spectrum of the input, S2(ν) is the output frequencyspectrum, s(t) is the temporal output signal and P1 is the average power arriving at thephotoelectric receiver. Parameter τ characterizes filter function H (ν).

10.5 Problems

1. Determine rise time τr for a simple RC circuit. Relate it to 3 dB bandwidth.2. Analyse noise in an avalanche photodiode (APD). Determine signal-to-noise ratio in

terms of gain of APD.3. Analyse signal limitations for an analogue transmission. Determine signal power in

terms of S/N ratio.

10.6 Projects

1. Conduct small-signal equivalent circuit analysis of a photodetector [1]. Write a MAT-LAB program. Determine frequency dependence of the small signal impedance of aphotodiode.

2. Analyse frequency response of a photodiode [17]. Determine the upper limit of thefrequency response of GaAs p-n junction photodiode assuming data provided in [17].


In Table 10.3 we provide a list of MATLAB files created for Chapter 10 and a shortdescription of each function.

259 Appendix 10A



10A.1 min power.m Determines minimum signal power10A.2 berQ.m Plots bit error rate versus Q

Listing 10A.1 Function min power.m. MATLAB function which determines minimumsignal power for a PIN photodetector for three values of temperature.

% File name: min_power.m

% Determines minimum signal power required to give S/N ratio of one

% for a PIN detector

clear all

h_Planck = 6.6261d-34; % Planck’s constant (J s)

c_light=299.80d6; % speed of light (m/s)

k_B= 1.3807d-23; % Boltzmann constant (J/K)

e=1.602d-19; % electron’s charge (C)

%

C_j = 1d-12; % junction capacitance (1pF)

eta = 1; % quantum efficiency

lambda = 0.85d-6; % wavelength (0.85 microns)

B = 1d6:1d5:1d10; % range of detector bandwidth

%

for T_eff = [500 1000 1500]

P_min=(2*h_Planck*c_light/(eta*e*lambda))*sqrt(2*pi*k_B*T_eff*C_j)*B;

loglog(B,P_min,’LineWidth’,1.5)

hold on

end

%

xlabel(’Detector bandwidth (Hz)’,’FontSize’,14);

ylabel(’Minimum signal power (W)’,’FontSize’,14);

text(1d9, 4d-7, ’500K’,’Fontsize’,14)

text(5d7, 4d-7, ’1500K’,’Fontsize’,14)


grid on

pause

close all

Listing 10A.2 Program berQ.m. MATLAB program which plots bit error rate versus Q.Takes one minute to execute.

% File name: berQ.m

% Plots bit error rate versus Q

syms x % defines symbolic variable


f=(1/sqrt(2*pi))*exp(-x^2/2); % defines function for plot

Q_i = 3.5; Q_f = 8; Q_step = 0.5; % defines range and step

yy = double(int(f,x,Q_i,inf));

for Q = Q_i+Q_step:Q_step:Q_f

y = double(int(f,x,Q,inf));

yy = [yy,y];

end

Qplot = Q_i:Q_step:Q_f; % defines Q for plot

semilogy(Qplot,yy,’LineWidth’,1.5)

xlabel(’Q’,’FontSize’,14);

ylabel(’Bit Error Rate’,’FontSize’,14);


grid on

pause

close all

References

[1] S. B. Alexander. Optical Communication Receiver Design. SPIE Optical EngineeringPress and Institution of Electrical Engineers, Bellingham, Washington, and London,1997.

[2] R. Ramaswami and K. N. Sivarajan. Optical Networks: A Practical Perspective.Morgan Kaufmann Publishers, San Francisco, 1998.

[3] J. C. Cartledge and G. S. Burley. The effect of laser chirping on lightwave systemperformance. J. Lightwave Technol., 7:568–73, 1989.

[4] M. M.-K. Liu. Principles and Applications of Optical Communications. Irwin,Chicago, 1996.

[5] T. V. Muoi. Receiver design of optical-fiber systems. In E. E. Bert Basch, ed., Optical-Fiber Transmission, pp. 375–425. Howard W. Sams & Co., Indianapolis, 1986.

[6] D. L. Rogers. Integrated optical receivers using MSM detectors. J. Lightwave Technol.,9:1635–8, 1991.

[7] J.-M. Liu. Photonic Devices. Cambridge University Press, Cambridge, 2005.[8] S. O. Kasap. Optoelectronics and Photonics: Principles and Practices. Prentice Hall,

Upper Saddle River, NJ, 2001.[9] J. C. Palais. Fiber Optic Communications. Fifth Edition. Prentice Hall, Upper Saddle

River, NJ, 2005.[10] G. Keiser. Optical Fiber Communications. Fourth Edition. McGraw-Hill, Boston,

2011.[11] J. Wilson and J. F. B. Hawkes. Optoelectronics. An Introduction. Prentice Hall, En-

glewood Cliffs, 1983.[12] G. Einarsson. Principles of Lightwave Communications. Wiley, Chichester, 1996.[13] G. Keiser. Local Area Networks. Second Edition. McGraw Hill, Boston, 2002.

261 References

[14] W. van Etten and J. van der Plaats. Fundamentals of Optical Fiber Communications.Prentice Hall, New York, 1991.

[15] A. E. Stevens. An integrate-and-dump receiver for fiber optic networks. PhD thesis,Columbia University, 1995.

[16] S. Geckeler. Optical Fiber Transmission System. Artech House, Inc., Norwood, MA,1987.

[17] A. Yariv and P. Yeh. Photonics. Optical Electronics in Modern Communications. SixthEdition. Oxford University Press, New York, Oxford, 2007.

11 Finite difference time domain (FDTD) formulation

The finite-difference time-domain (FDTD) method is a widely used numerical scheme forapproximate description of propagation of electromagnetic waves. It can be used to studysuch phenomena as pulse propagation in various media. It is a simple scheme and can bemastered relatively fast.

Definite reference on the subject is a book by Taflove and Hagness [1]. Several otherbooks can be cited: Kunz and Luebbers [2], Sullivan [3], Yu et al. [4], Elsherbeni and Demir[5]. Also, recently published books by Bondeson et al. [6] and by Garg [7] have instructivechapters on the FDTD method.

11.1 General formulation

To develop a general formulation, one starts from Maxwell’s equations which we cast in theform suitable for 1D, 2D and 3D discretizations. In the following, we assume that J = 0and ρ = 0; i.e there is no flow of current and no free charges. Assume also nonmagneticmaterial and therefore set μ = μ0. General Maxwell’s equations are

∇ × H = ∂D

∂t(11.1)

∇ × E = −μ0∂H

∂t(11.2)

Here E and H are, respectively, electric and magnetic fields and D is the electric displacementflux density which is related to electric field E. The relation between E and D will besummarized in later sections for more detailed models.

First we write the above equations in terms of components. The following cases aretypically considered:

1. 1D (one-dimensional) case: infinite medium in y-direction and z-direction and no changein those directions and therefore ∂

∂y = ∂∂z = 0.

2. 2D (two-dimensional) case: infinite medium in z-direction and no change in z-directionand therefore ∂

∂z = 0.3. 3D (three-dimensional) case: no restriction on variations in all three directions.

In this chapter we will not discuss the 3D case in detail since it requires significanthardware resources for typical calculations.

262

263 General formulation

11.1.1 Three-dimensional formulation

Use general mathematical formula

∇ × H =

∣∣∣∣∣∣∣ax ay az∂∂x

∂∂y

∂∂z

Hx Hy Hz

∣∣∣∣∣∣∣= ax

(∂Hz

∂y− ∂Hy

∂z

)− ay

(∂Hz

∂x− ∂Hx

∂z

)+ az

(∂Hy

∂x− ∂Hx

∂y

)By applying the above expression, Maxwell’s equations can be written as

∂Dx

∂t= ∂Hz

∂y− ∂Hy

∂z

−∂Dy

∂t= ∂Hz

∂x− ∂Hx

∂z∂Dz

∂t= ∂Hy

∂x− ∂Hx

∂y(11.3)

−μ0∂Hx

∂t= ∂Ez

∂y− ∂Ey

∂z

μ0∂Hy

∂t= ∂Ez

∂x− ∂Ex

∂z

−μ0∂Hz

∂t= ∂Ey

∂x− ∂Ex

∂y

These are general equations used as a starting point for 3D implementation, see e.g. [1].

11.1.2 Two-dimensional formulation

In the 2D model we assume that ∂∂z = 0 and evaluate

∇ × H =

∣∣∣∣∣∣∣ax ay az∂∂x

∂∂y 0

Hx Hy Hz

∣∣∣∣∣∣∣ = ax∂Hz

∂y− ay

∂Hz

∂x+ az

(∂Hy

∂x− ∂Hx

∂y

)

where ax, ay, az are (as before) unit vectors defining coordinate system. Using the aboveexpression, Maxwell’s equations in terms of components become

ax∂Hz

∂y− ay

∂Hz

∂x+ az

(∂Hy

∂x− ∂Hx

∂y

)= ax

∂Dx

∂t+ ay

∂Dy

∂t+ az

∂Dz

∂t

ax∂Ez

∂y− ay

∂Ez

∂x+ az

(∂Ey

∂x− ∂Ex

∂y

)= −μ0

(ax

∂Hx

∂t+ ay

∂Hy

∂t+ az

∂Hz

∂t

)The above equations are then usually split into two groups, see [1] and [8].


Group 1:

∂Dx

∂t= ∂Hz

∂y(11.4)

∂Dy

∂t= −∂Ez

∂x(11.5)

∂Hz

∂t= −μ0

(∂Ey

∂x− ∂Ex

∂y

)(11.6)

Group 2:

∂Hx

∂t= − 1

μ0

∂Hz

∂y(11.7)

∂Hy

∂t= 1

μ0

∂Ez

∂x(11.8)

∂Dz

∂t= ∂Hy

∂x− ∂Hx

∂y(11.9)

The first group is called T Ex mode by Taflove and TE to z polarization by Umashankar,and the group 2 is called T Mz mode by Taflove and TM to z polarization by Umashankar.

11.1.3 One-dimensional model

In one dimension, the medium extends to infinity in the y-direction and z-direction. There-fore ∂

∂y = ∂∂z = 0. Using the following result,

∇ × E =

∣∣∣∣∣∣∣ax ay az∂∂x 0 0

Hx Hy Hz

∣∣∣∣∣∣∣ = −ay∂Hz

∂x+ az

∂Hy

∂x

one can write Maxwell’s equations in two groups according to field vector components.First group is

∂Hy

∂t= 1

μ0

∂Ez

∂x(11.10)

∂Dz

∂t= ∂Hy

∂x(11.11)

Taflove [1] classifies this group as the x-directed, z-polarized TEM mode. It contains Dy

and Hy.The second group is

∂Hz

∂t= − 1

μ0

∂Ey

∂x(11.12)

∂Dy

∂t= −∂Hz

∂x(11.13)

This group is referred to as the x-directed, y-polarized TEM mode. It contains Dy and Hz.In the following, we will concentrate on group 2. The equations describe wave propagating

265 General formulation

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

time

ampl

itud

e

0 20 40 60 80 100−1

−0.5

0

0.5

1

time

ampl

itud

e

Fig. 11.1 Gaussian pulse (left), and modulated Gaussian pulse (right).

in the x-direction with electric field oriented in the y-direction and magnetic field orientedin the z-direction.

We finish this section with a discussion of the Gaussian pulse and modulated Gaussianpulse.

11.1.4 Gaussian pulse andmodulated Gaussian pulse

In practice one often analyses propagation of the Gaussian pulse or modulated Gaussianpulse. A Gaussian pulse centred at t0 is defined as

g(t) = exp

[− (t − t0)

2

w2

](11.14)

where w is the width of the pulse in space. Define also the modulation signal as

m(t) = sin (2π f0 · t) (11.15)

where f0 = c/λ0 = 1/T0. Here T0 is the period of the modulation signal m(t). A continuousGaussian pulse is created by multiplying Eqs. (11.14) and (11.15):

g(t) = sin (2π f0 · t) · exp

[− (t − t0)

2

w2

](11.16)

Its discretized version is

g(n) = sin (2π f0 · n · �t) · exp

[− (n · �t − t0)

2

w2

](11.17)

In Fig. 11.1 we plot Gaussian and modulated Gaussian pulses. The results were obtainedwith MATLAB code presented in Appendix, Listings 11A.1 and 11A.2.


11.2 One-dimensional Yee implementation without dispersion

In this section, we discuss the basic implementation of FDTD method in 1D. It is knownas the 1D-Yee algorithm since it was first proposed by Yee [9] for the 3D case. We alsointroduce the concept of a staggered grid [3], [7]. To illustrate basic principles of FDTDmethod, let us develop first the 1D lossless case.

11.2.1 Lossless case

Our formulation is based on the second group of Eqs. (11.12) and (11.13). We supplementthose equations with the relation between Dy and Ey, which for the free space is

Dy = ε0Ey (11.18)

The equations take the form

∂Hz

∂t= − 1

μ0

∂Ey

∂x(11.19)

∂Ey

∂t= − 1

ε0

∂Hz

∂x(11.20)

In order to introduce 1D Yee discretization, let us introduce the following notation:

Ey(i · �x, n · �t) ≡ (Ey

)n

i

Hz(i · �x, n · �t) ≡ (Hy

)n

i

To discretize Eqs. (11.19) and (11.20), we use central-difference approximation for spaceand time derivatives. They are second-order accurate in the space and time increments:

∂Ey

∂x(i · �x, n · �t) = (Ey)

ni+1/2 − (Ey)

ni−1/2

�x+ O(�x)2

∂Ey

∂t(i · �x, n · �t) = (Ey)

n+1/2i − (Ey)

n−1/2i

�t+ O(�t)2

In the Yee algorithm [9] E and H components are interleaved in the space lattice at intervalsof �x/2. Such an approximation is called the staggered grid approximation. They are alsointerleaved in time at intervals �t/2, see Fig. 11.2.

At time t = 0, values of Ey are placed at x = i · �x, with i = 0, 1, 2, . . . N . (Total N + 1components).

At time t = 12�t, values of Hz are placed at x = (i − 1

2 ) · �x, with i = 1, 2, . . . N . (TotalN components).

In general

(Ey)ni = Ey((i − 1) · �x, n · �t), i = 1, 2, . . . N + 1; n = 0, 1, 2 . . .

(Hz)ni = Hz

((i − 1

2

)· �x,

(n − 1

2

)· �t

), i = 1, 2, . . . N; n = 0, 1, 2 . . .

267 One-dimensional Yee implementation without dispersion

time

Hz Hz HzHz

Hz Hz HzHz

x

Ey Ey Ey Ey Ey

Ey Ey Ey Ey Ey

x = 0

t = 0

21

Δt

Δt

Δt

Δx 2Δx 3Δx 4Δx

23

Fig. 11.2 One-dimensional formulation of the FDTD method. Orientations ofHz andEy fields are shown.

Ey

Hz

n+1

n

n+2

i+-12i--12 i+1i-1i-2 i

n+-12

Fig. 11.3 Visual illustration of numerical dependencies in the 1D FDTD method.

Initial values (corresponding to n = 0), i.e. E(. . . , t = 0) and H (. . . , t = 0), in most casesare set to zero.

The discretized equations are

(Ey)n+1/2i − (Ey)

n−1/2i

�t= − 1

ε0

(Hz)ni+1/2 − (Hz)

ni−1/2

�x

(Hz)n+1i+1/2 − (Hz)

ni+1/2

�t= − 1

μ0

(Ey)n+1/2i+1 − (Ey)

n−1/2i+1

�x

The space-time interdependencies of Ey and Hz fields at different grid points are illustratedin Fig. 11.3. Here, squares represent Ey field and circles represent Hz field. One can observethat the value of a field at any point is determined by three previous values: two from twoneighbours of opposite field from the previous half time step and one from the same fieldat a single previous time step.


0 50 100 150 200−1

−0.5

0

0.5

1

Ey

FDTD cells

0 50 100 150 200−1

−0.5

0

0.5

1

Hz

FDTD cells

Fig. 11.4 FDTD simulation of a Gaussian pulse after several time steps.

Constants ε0 and μ0 appearing in these equations differ by several orders of magnitude.Therefore, one usually introduces a new scaled variable defined as [3]

Ey =√

ε0

μ0Ey

Scaled equations are

En+1/2y (i) = En−1/2

y (i) − 1√ε0μ0

�t

�x

[H n

z (i + 1/2) − H nz (i − 1/2)

]H n+1

z (i + 1/2) = H nz (i + 1/2) − 1√

ε0μ0

�t

�x

[En+1/2

y (i + 1) − En+1/2y (i)

]In our simulations, due to stability reasons we set

1√ε0μ0

�x

�t= 1

2(11.21)

The value of v = �x/�t is chosen to be the maximum phase velocity of the wave expectedin the medium. The above system has been implemented. The MATLAB program is shownin Appendix, Listing 11A.3. Two versions of MATLAB code have been produced: one ofwhich uses regular loop and one that takes advantage of parallel capabilities of MATLAB.The reader should analyse speed of operation for both versions.

The results of propagation of Gaussian pulse using that code are shown in Fig. 11.4.


11.2.2 Determination of cell size

In one dimension the cell size is just �x. In 2D it is a square and a cube in 3D. Thefundamental question is how to determine size of basic cell.

For a detailed discussion we refer to the book by Kunz and Luebbers [2]. For us, it isenough to notice that cell size should be much less than the smallest wavelength and typicalrule is to take it as λ/10. Such choice should provide accurate results.

11.2.3 Dispersion and stability

Here, we discuss some general properties of the wave equation. It was formulated before.Its discretized version is (here we use index r instead of i to avoid conflict with a symboli = √−1)

En+1r − 2En

r + En−1r

�t2= c2 En

r+1 − 2Enr + En

r−1

�x2(11.22)

As usual, n refers to time and r is the space index. Explicitly Enr = E(r · �z, n · �t). The

dispersion relation for Eq. (11.22) is found by assuming

Enr = ei(ω·n·�t−k·r·�x)

Substituting these assumptions into Eq. (11.22) gives[eiω·�t·(n+1) − 2eiω·�t·n + eiω·�t·(n−1)

]e−ik·r·�x

= c2�t2

�x2

[e−ik·�z·(r+1) − 2e−ik·r·�x + e−ik·�x·(r−1)

]eiω·�t·n

Dividing the above by exp[i (ω · n · �t − k · r · �x)]:

eiω·�t − 2 + e−iω·�t = c2�t2

�z2

(e−ik·�z − 2 + eik·�z

)The above can be written as a square:(

eiω·�t/2 − e−iω·�t/2)2 = c2�t2

�x2

(eik·�x/2 − e−ik·�x/2

)2

Finally, using trigonometric relation eiα−e−iα = 2i sin α, one obtains formula for numericaldispersion:

sin1

2ω · �t = ±c�t

�xsin

1

2k · �x (11.23)

The important case is that of linear dispersion, i.e. when ω = c · k. Eq. (11.23) then holdswhen �x = c�t. For such a linear case different frequency components propagate withthe same speed, just c. For nonlinear dispersion relations, different frequencies of a pulsenumerically propagate with different velocities and therefore pulse will change shape.

Introduce parameter

α = c�t

�z(11.24)


0 1 2 3 40

0.5

1

1.5

2

2.5

3

3.5

k Δ x

ωΔ

z /c

α = 1

α = 0.8

α = 0.2

Fig. 11.5 Illustration of numerical dispersion for various values of parameterα.

In Fig. 11.5 we showed numerical dispersions as described by Eq. (11.23) for several valuesof parameter α. MATLAB code is shown in Appendix, Listing 11A.4. Depending on thevalue of α, one can distinguish the following cases:

1. α = 1. In this case �t = �xc . Numerical dispersion is linear and given by ω = ±c·k. This

choice of time step �t is called the magic time step [10]. In this case wave propagatesexactly one cell (or �x) per time step in both x-directions.

2. α < 1. In this case �t < �xc . One observes numerical dispersion which increases when

α decreases.3. α > 1. In this case �t > �x

c . This inequality implies that numerical front propagatesfaster than physical speed (given by c). In this case, the scheme becomes unstable.

Because of its importance, we provide next a separate discussion of the stability criterion.

11.2.4 Stability criterion

Stability criterion imposes condition on �x�t ratio. It is known as Courant-Friedrichs-Levy

(CFL) condition [11]. It explains convergence of various schemes.For the 1D model, the CFL condition is illustrated in Fig. 11.6. Each row of points

represents the sampling points in space at a given instant of time (say n). The solution atpoint A is determined using previous values at the indicated points. In the figure we showcharacteristics which correspond to stable (left) and unstable (right) situations.

The stability criterion summarizes the physical fact that speed of numerical propagationshould not exceed the physical speed. A summary of stability criteria for various dimensionsis presented in Table 11.1 [10].


Table 11.1 Summary of stability criteria for various dimensions.

Dimensionality Criterion

1D medium v �t ≤ �x

2D medium v �t ≤(

1�x2 + 1

�y2

)−1/2

3D medium v �t ≤(

1�x2 + 1

�y2 + 1�z2

)−1/2

t

x

A

Δt

Δx

i i+1i-1

n

n-3

n-1

n-2

i-2 i+2

t

x

A

i i+1i-1

n

n-3

n-1

n-2

i-2 i+2

Fig. 11.6 Illustration of the CFL condition. For the parameters chosen within the dashed lines, the propagation is stable (leftfigure) and becomes unstable (right figure.)

11.2.5 One-dimensional model with losses

When we consider ohmic losses, Maxwell’s Eq. (11.1) is modified as

∇ × H = ∂D

∂t+ σE (11.25)

where σ is the conductivity.The discretization of Eqs. (11.25) and (11.2) with losses is done in a similar way as for

the lossless case. The resulting equations after scaling are

(Ey)n+1/2i − (Ey)

n−1/2i

�t= − 1

εr√

ε0μ0

(Hz)ni+1/2 − (Hz)

ni−1/2

�x

− σ

2εrε0

((Ey)

n+1/2i + (Ey)

n−1/2i

)(11.26)

and

(Hz)n+1i+1/2 − (Hz)

ni+1/2

�t= − 1

μ0

(Ey)n+1/2i+1 − (Ey)

n−1/2i

�x(11.27)


time

boundary

boundary point

direction oftravelling wave

xi = 0 i = 1/2 i = 1

tn = nΔt

tn+ _ = ( )Δt12

n+12_

Fig. 11.7 Illustration of propagating wave close to the left boundary.

When selecting the time step given by Eq. (11.21), one arrives at the following equations:

(Ey)n+1/2i − (Ey)

n−1/2i = − 1

2εr

((Hz)

ni+1/2 − (Hz)

ni−1/2

)− σ�t

2εrε0

((Ey)

n+1/2i + (Ey)

n−1/2i

)(Hz)

n+1i+1/2 − (Hz)

ni+1/2 = − 1

μ0

�t

�x

((Ey)

n+1/2i+1 − (Ey)

n−1/2i

)Solving the first equation for En+1/2

y (i) gives

(Ey)n+1/2i =

1 − σ�t2εrε0

1 + σ�t2εrε0

(Ey)n−1/2i − 1

2εr

(1 + σ�t

2εrε0

) ((Hz)ni+1/2 − (Hz)

ni−1/2

)(11.28)

Implementation and analysis of the above equations is left as a project.

11.3 Boundary conditions in 1D

11.3.1 Mur’s first-order absorbing boundary conditions (ABC)

Here, based on intuitive arguments we derive the so-called Mur’s first-order absorbingboundary conditions [12]. Consider plane wave φ(x, t) travelling left at the left boundary,see Fig. 11.7.

Reflections of this wave will be eliminated if [7]

φ(x, t) = φ(x − �x, t + �t) (11.29)

Perform Taylor expansion

φ(x − �x, t + �t) = φ(x, t) − ∂φ

∂x�x + ∂φ

∂t�t

273 Boundary conditions in 1D

Velocity of the propagating wave is

c = �x

�t

Substitute into Eq. (11.29) and have

φ(x, t) = φ(x, t) − ∂φ

∂xc�t + ∂φ

∂t�t

or∂φ

∂x= 1

c

∂φ

∂t(11.30)

It describes ABC for a plane wave incident from the right.Finite difference of Eq. (11.30) close to the left boundary is

φn+1/21 − φ

n+1/20

�x= 1

c

φn+11/2 − φn

1/2

�t(11.31)

The values at half grid points and half time steps are determined as

φn+1/2i = 1

2

(φn+1

i + φni

)φn

i+1/2 = 1

2

(φn

i+1 + φni

)Substitute the above into Eq. (11.31) and obtain an expression for φn+1

0 :

φn+10 = φn

1 + c�t − �x

c�t + �x

(φn+1

1 + φn0

)(11.32)

The formula (11.32) determines the new (updated) field value at the boundary node pointat x = 0.

The above expression is known as the first-order Mur’s absorbing boundary conditions(ABC). A similar derivation can be performed for right boundary. It is left as an exercise.

11.3.2 Second-order boundary conditions in 1D

We will provide a more rigorous discussion of the boundary conditions based on the workby Umashankar [8] and Taflove [10]. It allows for the introduction of the second-orderMur’s radiation boundary conditions.

We start with a 1D wave equation for any scalar component of the field

∂2U

∂x2− 1

c2

∂2U

∂t2= 0 (11.33)

Introduce operator L defined as

L = ∂2

∂x2− 1

c2

∂2

∂t2= D2

x − 1

c2D2

t

The wave equation can therefore be written as

LU = 0


0

Δx

U0 U1/2 U1 U2 UN-1 UNUN-1/2

xx1x2x1/2 xN-1 xN-1/2 xN

Fig. 11.8 Illustration of derivation of ABC in one dimension.

The above operator can be factored as

LU = L+L−U = 0

where L+ = Dx + 1c Dt and L− = Dx − 1

c Dt .Suppose that 1D computational region extends from 0 to h; i.e. one has 0 ≤ x ≤ h.

Engquist and Majda [13] showed that at x = 0 the application of an operator L− to Uexactly absorbs a plane wave propagating towards the boundary. A similar relation holdsfor an operator L+ operating at x = h.

In light of the above, the PDE that can be numerically implemented as a first-orderaccurate ABC at x = 0 grid boundary is

L−U =(

Dx − 1

cDt

)u = 0

Multiplying the above by an operator Dt , one finds

Dt

(Dx − 1

cDt

)U = 0

In the full form∂2U

∂t∂x− 1

c

∂2U

∂t2= 0 (11.34)

which is a second-order ABC at x = 0. For the above equation original numerical schemeshave been proposed by Mur [12]. Here, we followed an improved scheme as described byTaflove [10].

Let U represent a Cartesian component of E or H located on the Yee grid, see Fig. 11.8.We use central differences expansion around an auxiliary grid point x = 1/2 and timetn = n · �t. The derivatives are approximated as follows:

∂2U

∂t∂x

∣∣∣∣n1/2

= 1

2�t

[∂U

∂x

∣∣∣∣n+1

1/2

− ∂U

∂x

∣∣∣∣n−1

1/2

]

= 1

2�t

[U n+1

1 − U n+10

�x− U n−1

1 − U n−10

�x

]and

∂2U

∂t2

∣∣∣∣n1/2

= 1

2

[∂2U

∂t2

∣∣∣∣n0

+ ∂2U

∂t2

∣∣∣∣n1

]

= 1

2

[U n+1

0 − 2U n0 + U n−1

0

�t2+ U n+1

1 − 2U n1 + U n−1

1

�t2

]

275 Two-dimensional Yee implementation without dispersion

Substitution of the above results into Eq. (11.34) and solving for U n+10 gives

U n+10 = U n−1

0 + c�t − �x

c�t + �x

(U n+1

1 + U n−10

)+ 2�x

c�t + �x

(U n

1 + U n0

)(11.35)

That expression determines the value of the field U (x, t) at the next time tn+1 = (n+1) ·�tfor x = 0.

In exactly the same way one obtains the updated equation at the right boundary. TheABC wave equation is

∂2U

∂t∂x+ 1

c

∂2U

∂t2= 0 (11.36)

The derivatives are approximated as follows:

∂2U

∂t∂x

∣∣∣∣nN−1/2

= 1

2�t

[∂U

∂x

∣∣∣∣n+1

N−1/2

− ∂U

∂x

∣∣∣∣n−1

N−1/2

]

= 1

2�t

[U n+1

N − U n+1N−1

�x− U n−1

N − U n−1N−1

�x

]and

∂2U

∂t2

∣∣∣∣nN−1/2

= 1

2

[∂2U

∂t2

∣∣∣∣nN

+ ∂2U

∂t2

∣∣∣∣nN−1

]

= 1

2

[U n+1

N − 2U nN + U n−1

N

�t2+ U n+1

N−1 − 2U nN−1 + U n−1

N−1

�t2

]Substituting the last result into Eq. (11.36) and solving for U n+1

0 gives

U n+1N = U n−1

N−1 + c�t − �x

c�t + �x

(U n+1

N−1 + U n−1N

)+ 2�x

c�t + �x

(U n

N + U nN−1

)(11.37)

11.4 Two-dimensional Yee implementation without dispersion

We assume medium with the constitutive relation of the form

Dy = ε0εrEy

Using the above constitutive relation, from the group 2 (Eqs. 11.8) (also called T Mz mode)of Maxwell’s equations one has (ε = ε0εr)

∂Ez

∂t= 1

ε

(∂Hy

∂x− ∂Hx

∂y

)∂Hx

∂t= − 1

μ0

∂Hz

∂y

∂Hy

∂t= 1

μ0

∂Ez

∂x


ii-1 i+1i-1/2 i+1/2

j

j+1

j+1/2

j-1

j-1/2

Ez

Ez

EzEz

x

cell

y

Hy Hy

HxHx

Hx

Hx

HxHx

Δx

Δy

Fig. 11.9 Yee mesh in 2D for theTMz mode.

For 2D discretization we introduce the Yee mesh, shown in Fig. 11.9. A Yee cell is formedaround point (i, j). The electric field distribution within each cell is assumed to be constantand the z-component Ez of electric field is defined at the midpoint of each cell. We alsointroduce (i, j) = (i�x, j�y) and Fn(i, j) = F (i�x, j�y, n�t). Maxwell’s equations areapproximated as

En+1z (i, j) − En

z (i, j)

�t= 1

ε0εr(i, j)

H n+1/2y (i + 1/2, j) − H n+1/2

y (i − 1/2, j)

�x(11.38)

− 1

ε0εr(i, j)

H n+1/2x (i, j + 1/2) − H n+1/2

x (i, j − 1/2)

�y

Hn+1/2x (i, j + 1/2) − H n−1/2

x (i, j + 1/2)

�t= − 1

μ0

Enz (i, j + 1) − En

z (i, j)

�y(11.39)

H n+1/2y (i + 1/2, j) − H n−1/2

y (i + 1/2, j)

�t= 1

μ0

Enz (i + 1, j) − En

z (i, j)

�x(11.40)

From the above equations one determines the time-stepping numerical algorithm for theinterior region. For the same reason as in the 1D case, we need to perform scaling. Asbefore we introduce new scaled variable Ez as follows:

Ez =√

ε0

μ0Ez (11.41)

An algorithm for scaled variables is

En+1z (i, j) = En

z (i, j) + �t

ε0εr(i, j)

1√ε0μ0

{H n+1/2

y (i + 1/2, j) − H n+1/2y (i − 1/2, j)

�x

− H n+1/2x (i, j + 1/2) − H n+1/2

x (i, j − 1/2)

�y

}(11.42)

277 Absorbing boundary conditions (ABC) in 2D

020

4060

020

40600

0.5

1

x

time = 20

y

Ez

020

4060

020

4060

0

0.5

1

x

time = 30

y

Ez

020

4060

020

40600

0.5

1

x

time = 40

y

Ez

020

4060

020

4060

0

0.5

1

x

time = 60

y

Ez

Fig. 11.10 Propagation of a Gaussian pulse initiated in the middle and travelling outwards. Four values of time are shown.

H n+1/2x (i, j + 1/2) = H n−1/2

x (i, j + 1/2) − �t

�y

1√ε0μ0

[En

z (i, j + 1) − Enz (i, j)

](11.43)

H n+1/2y (i + 1/2, j) = H n−1/2

y (i + 1/2, j) − �t

�y

1√ε0μ0

[En

z (i + 1, j) − Enz (i, j)

](11.44)

The above equations were implemented in MATLAB for the propagation of Gaussianpulse. A MATLAB program implementing 2D propagation of the Gaussian pulse is shownin Appendix, Listing 11A.6.

Propagation of the Gaussian pulse initiated in the middle and travelling outwards isshown in Fig. 11.10. We illustrate pulse shape at four values of time. One can then observeits spread in time.

11.5 Absorbing boundary conditions (ABC) in 2D

Absorbing boundary conditions (ABC) in 2D are derived here. We follow the methodologybased on the wave equation used in the 1D case [7] and [10]. The two-dimensional wave


h

hx

y

0

Fig. 11.11 Rectangular region used to illustrate derivations of ABC.

equation for scalar component U is

1

c2

∂2U

∂t2= ∂2U

∂x2+ ∂2U

∂y2

In a rectangular coordinate system, the plane-wave time-harmonic solution to wave equationis

U (x, y, t) = ei(ωt−kxx−kyy)

Substituting into a wave equation gives the condition

k2x + k2

y = ω2

c2≡ k2

Solution for kx is therefore

kx = ±(

k2 − k2y

)1/2

= ±k

(1 − k2

y

k2

)1/2

� ±k

(1 − 1

2

k2y

k2

)The last result can be written as

kkx = ±k2 ∓ 1

2k2

y (11.45)

From the above relation, we can reconstruct partial differential equations that can be nu-merically implemented as a second-order accurate ABC. First observe that for a rectangularregion shown in Fig. 11.11 near x = 0 boundary, the left travelling wave is described by thefollowing equation:

1

c

∂2U

∂x∂t= 1

c2

∂2U

∂t2− 1

2

∂2U

∂y2(11.46)

Analogously, at the x = h boundary

1

c

∂2U

∂x∂t= − 1

c2

∂2U

∂t2+ 1

2

∂2U

∂y2

279 Absorbing boundary conditions (ABC) in 2D

y

xx = 0

U0,j

Uy = ( j+1)Δy

y = ( j–1)Δyx = Δx

y = jΔy

0,j+1 U1,j+1

U1,j-1U0,j-1

U1,j

U1/2,j

Fig. 11.12 Numerical boundary conditions.

To obtain numerical boundary conditions, we approximate the above derivatives around thecentral point ( 1

2 , j), see Fig. 11.12.One obtains

∂2U n1/2, j

∂x∂t= 1

2�t

{∂U n+1

1/2, j

∂x−

∂U n−11/2, j

∂x

}

=U n+1

1, j − U n+10, j

2�t�x−

U n−11, j − U n−1

0, j

2�t�x

The second-order derivative with respect to time t is implemented as an average of timederivatives at the adjacement points (0, j) and (1, j) as

∂2U n1/2, j

∂t2= 1

2

{∂2U n

0, j

∂t2+ ∂2U n

1, j

∂t2

}

=U n+1

0, j − 2U n0, j + U n−1

0, j

2�t2+

U n+11, j − 2U n

1, j + U n−11, j

2�t2

Also, as an average of the y derivatives we implement

∂2U n1/2, j

∂y2= 1

2

{∂2U n

0, j

∂y2+ ∂2U n

1, j

∂y2

}

= U n0, j+1 − 2U n

0, j + U n0, j−1

2�y2+ U n

1, j+1 − 2U n1, j + U n

1, j−1

2�y2

Substitute the above results into Eq. (11.46) and have

U n+11, j − U n+1

0, j

2�t�x−

U n−11, j − U n−1

0, j

2�t�x

= − 1

c22�t2

(U n+1

0, j − 2U n0, j + U n−1

0, j + U n+11, j − 2U n

1, j + U n−11, j

)(11.47)

= − 1

4�y2

(U n

0, j+1 − 2U n0, j + U n

0, j−1 + U n1, j+1 − 2U n

1, j + U n1, j−1

)From the above we can find the time-stepping algorithm for the component of U alongx = 0 boundary, i.e. U n+1

0, j .


11.6 Dispersion

Dispersion exists when medium properties are frequency-dependent. Examples includedielectric constant and/or conductivity which can vary with frequency. Generally, the fol-lowing situations exist [1]:

• linear dispersion• nonlinearity• nonlinear dispersion• gain.

Here, we will only summarize the most important models of material dispersion andbriefly outline one of the possible numerical approaches to dispersion.

11.6.1 Material dispersion

Usually, the dependencies are established in frequency domain. The defining equation is

D(ω) = ε0εr(ω)E(ω)

The physics is contained in εr(ω). We assume non-conducting media. Main cases are:

• Debye medium (first order)

ε(ω) = ε∞ + εs − ε∞1 + jωt0

(11.48)

• Lorentz medium

εr(ω) = εr + ε1

1 + 2 jδ0

(ωω0

)−(

ωω0

)2

Specific analytical and numerical work with the inclusion of dispersion is left as a project.

11.7 Problems

1. Derive an expression for the first-order Mur’s boundary conditions at the rightinterface.

2. Determine the residual reflection produced by Mur’s first-order ABC. Assume theexistence of the following wave at the boundary:

φ(x, t) = exp (iωt) · [exp (ikx) + R exp (−ikx)] (11.49)

Analyse reflection coefficient R.

281 Appendix 11A



11A.1 gauss.m Plots Gaussian pulse11A.2 gauss modul.m Plots modulated Gaussian pulse11A.3 f dtd f ree.m Propagation of Gaussian pulse in a free space11A.4 num disp.m Illustrates numerical dispersion11A.5 gm f .m Animates Gaussian pulse in a free space (not used in text)11A.6 f dtd 2D.m Propagation of Gaussian pulse in 2D

11.8 Projects

1. Implement a 1D FDTD approach with losses. Analyse propagation of Gaussian andsuper-Gaussian pulses.

2. Implement boundary conditions for 1D and 2D problems.3. Use a convolution integral approach which relates D and E:

D(t) = ε∞ε0E(t) + ε0

∫ t

0E(t − t ′)χ(t ′)dt ′ (11.50)

to develop discretization scheme for the first-order Debye medium as described byEq. (11.48). Implement in MATLAB the resulting numerical scheme. For more details,consult the book by Kunz and Lubbers [2].

4. Conduct similar analysis for the first-order Drude dispersion; i.e.

ε(ω) = 1 + ω2p

ω ( jνc − ω)(11.51)

where ωp is the plasma frequency and νc is the collision frequency.



Listing 11A.1 Program gauss.m. MATLAB program which creates Gaussian pulse.

% File name: gauss.m

% Plots Gaussian pulse

clear all;

%




%

N = 300.0;

time = linspace(0,50,N);

pulse = exp(-0.5*((t_zero - time)/width).^2); % Gaussian pulse

plot(time,pulse,’LineWidth’,1.5)

xlabel(’time’,’FontSize’,14);

ylabel(’amplitude’,’FontSize’,14);


pause

close all

Listing 11A.2 Program gauss modul.m. MATLAB program which creates a modulatedGaussian pulse.

% File name: gauss_modul.m

% Plots modulated Gaussian pulse

alpha = 0.6;

mu_0=4*pi*1e-7; e_0=8.854e-12; % fundamental constants

c=1/sqrt(mu_0*e_0); % velocity of light

dx = 4.2e-2; % step in space

dt=alpha*dx/c; % time step

f_mod = 2e9; % modulating frequency (2 GHz)

t_0 = 3e-09; % peak position of Gaussian pulse in time

T = t_0/3; % pulse width

N_x = 400;

spec = [0:N_x];

for n = 1:100

y(n) = exp(-(((n-1)*dt-t_0)/T).^2).*cos(2*pi*f_mod*((n-1)*dt-t_0));

end

plot(y,’LineWidth’,1.5)



pause

close all

Listing 11A.3 Function fdtd free.m MATLAB program which describes propagation ofa Gaussian pulse in a free space using the FDTD method.

% File name: fdtd_free.m

% Plots E and H components of Gaussian pulse in a free space

% after some number of time steps using FDTD method

% No animation

clear all;

N_x = 200; % number of x-cells

283 Appendix 11A

N_time = 100; % number of time steps

time = 0.0;

E_y = zeros(N_x,1); % Initialize fields to zero at all points

H_z = zeros(N_x,1);


width = 12; % width of the incident pulse

tic; % start timer

for istep = 1:N_time % time loop

time = time + 1;

pulse = exp(-0.5*((t_zero - time)/width)^2); % Gaussian pulse

for i = 2:(N_x)

E_y(120) = pulse; % location of initial pulse

E_y(i) = E_y(i) - 0.5*(H_z(i) - H_z(i-1));

end

%

for i = 1:(N_x -1)

H_z(i) = H_z(i) - 0.5*(E_y(i+1) - E_y(i));

end

end

toc % measure elapsed time since last call to tic

subplot(2,1,1); plot(E_y)

ylabel(’E_y’); xlabel(’FDTD cells’)

axis([0 200 -1 1])

subplot(2,1,2); plot(H_z)

ylabel(’H_z’); xlabel(’FDTD cells’)

axis([0 200 -1 1])

pause

close all

Listing 11A.4 Function num disp.m. MATLAB program used to produce Fig. 11.5 toillustrate numerical dispersion.

% File name: num_disp.m

% Illustrates numerical dispersion


x = linspace(0,pi,N_max);

hold on

for alpha = [1.0 0.8 0.2]

y = 2*asin(alpha*sin(x/2));

plot(x,y,’LineWidth’,1.5)

end

xlabel(’k \Delta z’,’FontSize’,14)

ylabel(’\omega \Delta x /c’,’FontSize’,14)



text(2.5, 3, ’\alpha = 1’,’Fontsize’,12)

text(2.5, 2, ’\alpha = 0.8’,’Fontsize’,12)

text(2.5, 0.5, ’\alpha = 0.2’,’Fontsize’,12)

pause

close all

Listing 11A.5 Function gmf.m. MATLAB program which generates and animates prop-agation of a Gaussian pulse in a free space.

% File name: gmf.m

% gauss, modulated, free space

% Animation of propagation of Gaussian pulse in a free space

% without boundary conditions

% Program not used in the main text

clear all

alpha = 0.5; % alpha = c*dt/dx

mu0=4*pi*1e-7; % magnetic permeability

e0=8.854e-12; % electric permittivity

c=1/sqrt(mu0*e0); % velocity of light in a vacuum

dx = 4.2e-2; % space step

dt=alpha*dx/c; % time step

f_mod = 1e9; % modulating frequency

lambda = c/f_mod; % wavelength

x_end = 400;

t_0 = 3e-09; % peak position of Gaussian pulse in time

dxl = dx/lambda;

index=[0:x_end];

e=zeros(length(index),1); h=e; e1=e; h0=e;

ha = plot(e); % get the handle to the plot

set(ha,’EraseMode’,’xor’) % for simulation of wave propagation

%

t_end = 500; % number of time steps

for n = 1:t_end

for i=1:400 % simulate for h-nodes

h(i) = h(i) - (e(i+1)-e(i))*dt/dx/mu0;

end

for i=2:400 % simulate for e-nodes

e(i) = e(i) - (h(i)-h(i-1))*dt/dx/e0;

end

% launch Gaussian or modulated Gaussian pulse

T = t_0/3; % specify pulse width

y(n) = exp(-(((n-1)*dt-t_0)/T).^2).*cos(2*pi*f_mod*((n-1)*dt-t_0));

% y(n) = exp(-(((n-1)*dt-t_0)/T).^2);

e(1)= y(n); % Gauss pulse starts here

drawnow; %refresh data and plot the current condition of the pulse

285 Appendix 11A

set(ha,’YData’,(e));

end

pause

close all

Listing 11A.6 Function fdtd 2D.m. MATLAB program which describes propagation ofa Gaussian pulse in 2D.

% File name: fdtd_2D.m

% Program for finite-difference time-domain method in 2D

% Propagation of Gaussian pulse

clear all;

N_x = 60; % number of steps in x-direction

N_y = 60; % number of steps in y-direction

N_steps = 60; % number of time steps

%

N_x_middle = N_x/2.0; % middle point; location of pulse

N_y_middle = N_y/2.0; %

%

delta_x = 0.01; % cell size

delta_t = delta_x/6e8; % time step

epsilon = 8.8e-12;

%

% Initialization

E_z = zeros(N_x,N_y); % Initialize field to zero at all points

H_x = zeros(N_x,N_y);

H_y = zeros(N_x,N_y);

%



time = 0.0;

%

for istep = 1:N_steps % main loop

time = time + 1;

% calculate E_z field

for i = 2:N_x

for j = 2:N_y

E_z(i,j)=E_z(i,j)+0.5*(H_y(i,j)-H_y(i-1,j)-H_x(i,j)+H_x(i,j-1));

end

end

% Gaussian pulse in the middle

pulse = exp(-0.5*((t_zero - time)/width)^2); % Gaussian pulse

E_z(N_x_middle,N_y_middle) = pulse;

%

% Calculate H_x field

for i = 1:(N_x-1)

for j = 1:(N_y-1)


H_x(i,j) = H_x(i,j) - 0.5*(E_z(i,j+1) - E_z(i,j));

end

end

%

% Calculate H_y field

for i = 1:(N_x-1)

for j = 1:(N_y-1)

H_y(i,j) = H_y(i,j) + 0.5*(E_z(i+1,j) - E_z(i,j));

end

end

end

%----------- Plotting------------------------------

[x,y] = meshgrid(0:1:N_x-1,0:1:N_y-1);

mesh(x,y,E_z);

xlabel(’\bfx’,’FontSize’,14);

ylabel(’\bfy’,’FontSize’,14);

zlabel(’\bfE_z’,’FontSize’,14);


axis([0 60 0 60 0 1])

text(50,40,0.5, ’time = 60’, ’Fontsize’, 14)

pause

close all

References

[1] A. Taflove and S. C. Hagness. Computational Electrodynamics. The Finite-DifferenceTime-Domain Method. Artech House, Boston, 2005.

[2] K. S. Kunz and R. J. Luebbers. The Finite Difference Time Domain Method forElectromagnetics. CRC Press, Boca Raton, 1993.

[3] D. M. Sullivan. Electromagnetic Simulation using the FDTD Method. IEEE Press,New York, 2000.

[4] W. Yu, X. Yang, Y. Liu, and R. Mittra. Electromagnetic Simulation Techniques Basedon the FDTD Method. Wiley, Hoboken, NJ, 2009.

[5] A. Z. Elsherbeni and V. Demir. The Finite-Difference Time-Domain Method for Elec-tromagnetics with Matlab Simulations. Scitech Publishing, Inc., Raleigh, NC, 2009.

[6] A. Bondeson, Th. Rylander, and P. Ingelstroem. Computational Electromagnetics.Springer, New York, 2005.

[7] R. Garg. Analytical and Computational Methods in Electromagnetics. Artech House,Boston, 2008.

[8] K. R. Umashankar. Finite-difference time domain method. In S. M. Rao, ed., TimeDomain Electromagnetics, pages 151–235. Academic Press, San Diego, 1999.

[9] K. S. Yee. Numerical solution of initial boundary value problems involving Maxwell’sequations in isotropic media. IEEE Trans. Antennas and Propag., 17:585, 1966.

287 References

[10] A. Taflove. Computational Electrodynamics. The Finite-Difference Time-DomainMethod. Artech House, Boston, 1995.

[11] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery. Numerical Recipesin Fortran 90. The Art of Scientific Computing. Second Edition. Cambridge UniversityPress, Cambridge, 1992.

[12] G. Mur. Absorbing boundary conditions for the finite-difference approximation of thetime-domain electromagnetic-field equations. IEEE Trans. Electromagn. Compat.,23:377–82, 1981.

[13] B. Engquist and A. Majda. Absorbing boundary conditions for the numerical simula-tion of waves. Math. Comp., 31:629–51, 1977.

12 Beam propagationmethod (BPM)

In this chapter we summarize the beam propagation method (BPM). The method is widelyused for the numerical solution of the Helmholtz equation and also for the numericalsolution to the nonlinear Schroedinger equation (to be discussed in Chapter 15 dealingwith solitons). It is the most powerful technique for studying the propagation of light inintegrated optics. The method was originally introduced by Feit and Fleck in the late 1970s[1]. The BPM was initially based on FFT algorithm. Later on it has been extended tofinite-difference based BPM schemes (FD-BPM) and finite-element BPM (FE-BPM) andmany others. The main characteristics [2] of FD-BPM are the ability to simulate structureswith large index discontinuity, less memory and time consumption in modelling complexstructures, the possibility to incorporate wide-angle and full vector algorithms and theability to incorporate transparent boundary conditions.

There is a large number of algorithms available in the literature and almost all of themare based on the concept of a propagator. Propagators are mathematical ‘objects’ whichpropagate fields from one space coordinate to another. An example of the system wherepropagation takes place, known as an optical waveguide, is shown in Fig. 12.1. It splitsinput optical signal into two arms, see also Fig. 12.2. The role of BPM is to determine fieldprofile along the waveguide knowing the distribution of refractive index over the wholewaveguide.

There is a huge amount of literature on the principles of BPM and its applications inintegrated photonics. We recommend books by Kawano and Kitoh [3], Pollock and Lipson[4], Okamoto [5] and Lifante [6]. We start our discussion by illustrating the principles ofBPM in the paraxial approximation.

12.1 Paraxial formulation

12.1.1 Introduction

We start this section by outlining the simplest version of BPM, where one assumes scalarelectric field and paraxial approximations which restrict its applicability to the fields prop-agating at small angles with respect to the axis of the waveguide (guiding axis), whichwe define as the z axis. The geometry of the waveguide is determined by refractive indexn(x, y, z). The method consists of the repeated propagation of the electric field from aperpendicular plane at a given position along the waveguide to the next parallel plane. To

288

289 Paraxial formulation

Fig. 12.1 Example of a planar waveguide. Here we show the beam splitter.

Fig. 12.2 Schematical illustration of a typical problem solved by BPM. Knowing the input electric field on the left, determinefields at the output (shown by dotted lines on the right).

illustrate the method, one starts from the wave equation for a monochromatic wave:

∂2E

∂x2+ ∂2E

∂y2+ ∂2E

∂z2+ k2(x, y, z)E = 0 (12.1)

Here, the spatially dependent wavenumber is given by k(x, y, z) = k0n(x, y, z) and k0 =2π/λ is the wavenumber in free space. The main assumption is that the phase variation dueto propagation along the z axis represents the fastest variation in the field E. This variationis exemplified by introducing slowly varying field u and expressing E as

E(x, y, z) = u(x, y, z)e− jβz (12.2)

where β is a constant that represents the characteristic propagation wave vector, β = n0ω/c.Here n0 is a reference refractive index which can be, for example, the refractive index ofthe substrate or cladding. β represents the average phase variation of the field E and it isknown as propagation constant. Substituting Eq. (12.2) into Eq. (12.1) yields the equationequivalent to the exact Helmholtz for the slowly varying field

−∂2u

∂z2+ 2 jβ

∂u

∂z= ∂2u

∂x2+ ∂2u

∂y2+ (k2 − β2)u (12.3)

At this stage one assumes the slow variation of optical field in the propagation direction(SVEA or slowly varying envelope approximation), which requires∣∣∣∣∂2u

∂z2

∣∣∣∣ �∣∣∣∣2β

∂u

∂z

∣∣∣∣ (12.4)

290 Beam propagation method (BPM)

This approximation allows us to ignore the first term on the left hand side of Eq. (12.3)with respect to the second one. It is also known as Fresnel approximation. Eq. (12.3) thusreduces to

2iβ∂u

∂z= ∂2u

∂x2+ ∂2u

∂y2+ (k2 − β2)u (12.5)

which is known as a Fresnel or paraxial equation. It is a starting point for the descriptionof the evolution of electric (or magnetic) field in inhomogeneous medium, for example anoptical waveguide. It does not describe polarization effects. It determines the evolution ofthe field for z > 0 given an input u(x, y, z = 0).

The above equation can be expressed as

2 jβ∂u

∂z= Du + W u (12.6)

with the operators on the right hand side being

D = 1

2 jβ

(∂2

∂x2+ ∂2

∂y2

)(12.7)

W = 1

2 jβ

(k2 − β2

)(12.8)

Operator D represents free-space propagation (diffraction) and operator W describes guid-ing effects.

Assuming that operators are z-independent, the solution is symbolically written as

u(x, y, z + �z) = e(D+W )�zu(x, y, z)

Use the Baker-Hausdorf lemma [7]

eAeB = eA+Be12 [A,B]

given that A and B each commutes with [A, B]. Using it, we can approximate

u(x, y, z + �z) ≈ eD�zeW �zu(x, y, z)

From the above, it follows that actions due to both operators can be considered indepen-dently.

The action of operator D is better understood in the spectral domain. It will be de-scribed in the next section. The second operator W describes the effect of propagationin the presence of medium inhomogeneities and its action is incorporated in the spatialdomain.

12.1.2 Operators D andW

Consider an equation containing only operator D:

2 jβ∂u(x, y, z)

∂z=(

∂2

∂x2+ ∂2

∂y2

)u(x, y, z) (12.9)

291 Paraxial formulation

Define 2D continuous Fourier transform as

u(kx, ky, z) =+∞∫

−∞u(x, y, z)e− j(kxx+kyy)dxdy ≡ Fx,y {u(x, y, z)}

and its inverse

u(x, y, z) =+∞∫

−∞u(kx, ky, z)e j(kxx+kyy)dkxdkyF−1

x,y {u(x, y, z)}

Apply the 2D FT to Eq. (12.9) and have for Fourier components

2 jβ∂

∂zu(x, y, z) = −(k2

x + k2y )u(x, y, z)

Integrate the above from z to z + �z and have

u(x, y, z + �z) = e− 12 jβ (k2

x +k2y )�zu(x, y, z)

≡ H (kx, ky,�z)u(x, y, z) (12.10)

To analyse operator W , introduce the relation

β = k0ne f f (12.11)

Write also refractive index n as

n = ne f f + �n (12.12)

Substituting the above into Eq. (12.8) gives an expression for operator W to the first orderin �n

W = − jk0�n (12.13)

12.1.3 The implementation using the Fourier transform split-step method

A general scheme of FD-BPM is described by the propagator U , which we use to advancefield as

Et (z + �z) = U (�z)Et (z)

PropagatorU can take many forms depending on the chosen BPM technique. Its operationis illustrated in Fig. 12.3. In practice two versions of BPM are used: 1 + 1 FD-BPM and1 + 2 FD-BPM. The nomenclature refers to one dimension along the propagation direction(usually the z-axis) and one or two perpendicular dimensions (along x-axis (1D case) orin x-y plane (2D case)). Finite difference discretization in the 1 + 1 and 1 + 2 cases isillustrated in Fig. 12.4.

We illustrate the application of Fourier transform split-step BPM for the propagation ofa two-dimensional Gaussian pulse. The guiding structure is divided into a large number ofsegments. During each segment of length �z, the optical pulse is propagated as shown in


zz + Δz

Fig. 12.3 Illustration of the BPM scheme. Field profile is propagated from the position within one plane to a position withinanother plane along the z-axis.

XX X X

j

i-1

i-1

i

i

i+1

i+1

j+1

j-1

x

1 + 2 FD-BPM

1 + 1 FD-BPM

y

Fig. 12.4 Illustration of finite-difference discretization for 1D and 2D BPM.

the flow diagram, see Fig. 12.5. The accompanying MATLAB script is shown in Appendix,Listing 12B.1. The results of a 3D view of pulse before propagation and after propagationare shown in Fig. 12.6. One can observe spread of pulse width as expected.

12.2 General theory

12.2.1 Introduction

Polarization effects are not included in the previous version of BPM. When one wants todescribe polarization effects, the vector wave equation must replace the scalar Helmholtzequation used earlier. In this section we will outline a general approach to systematicallyhandle those effects. General formulation is based on an approach established by W. P.Huang’s group [8], [9]. As usual, one starts from Maxwell’s equations in the frequency

293 General theory

Initial profile u(x,y,z = 0)

Finished?

Yes

No

u(k ,k ,z) = F {u(x,y,z)}x y xy~

u(k ,k ,z + z) = H(k ,k z) u(k ,k ,z)x y x y x yΔ Δ,~ ~ˆ

u'(x,y,z + z) = F {u(k ,k ,z+ z)}Δ Δxy x y~-1

u(x,y,z + z) = e u '(x,y,z + z)Δ ΔSΔz^

Fig. 12.5 Flow diagram for the BPM split-step method.

domain:

∇ × E = − jωμ0H (12.14)

∇ × H = jωε0n2E (12.15)

∇ · n2E = 0 (12.16)

where we have introduced refractive index n as ε = ε0n2. We separate all quantities intolongitudinal (along z-axis) and transversal components, as follows:

∇ = ∇t + z∂

∂z, E = Et + zEz, H = Ht + zHz

where

∇t =[

∂

∂x,

∂

∂y, 0

], Et = [Ex, Ey, 0], Ht = [

Hx, Hy, 0]


−5

0

5

−5

0

50

0.2

0.4

0.6

0.8

1

−5

0

5

−5

0

50

0.2

0.4

0.6

0.8

Fig. 12.6 Initial Gaussian pulse (top). Pulse after propagation (bottom).

are the transversal components and z = [0, 0, 1] is the unit vector along z-axis. Eq. (12.16)can be rearranged as follows:

∇ · n2E = 0

295 General theory

or

∇t · n2Et + ∂

∂zn2Ez = 0

or

∇t · n2Et + Ez∂n2

∂z+ n2 ∂Ez

∂z= 0

∂

∂xn2Ex + ∂

∂yn2Ey + ∂

∂zn2Ez = 0

The z-derivative is

∂Ez

∂z= − 1

n2Ez

∂n2

∂z− 1

n2∇t · n2Et (12.17)

If the refractive index varies slowly along z-axis, we can set ∂n2

∂z ≈ 0 (z-invariant structures[9]) and obtain

∂Ez

∂z≈ − 1

n2∇t · n2Et

= − 1

n2

∂

∂xn2Ex − 1

n2

∂

∂yn2Ey (12.18)

The above relation is exact for z-invariant systems, i.e. when there is no change of refractiveindex n2 along z-axis.

The wave equation is derived by taking ∇ × · · · operation on Eq. (12.14). One finds

∇ × ∇ × E = − jωμ0∇ × H = ω2μ0ε0n2E (12.19)

In the last step we have used Eq. (12.15). Next, we use the following general relation whichholds for arbitrary vector

∇ × ∇ × E = ∇ (∇ · E) − ∇2E

Applying it to expression (12.19), we obtain wave equation

−∇2E + ∇ (∇ · E) = n2k20E (12.20)

where ω2μ0ε0 = k20 .

Transversal components of the wave equation (12.20) are

−∇2Ex + ∂

∂x(∇ · E) = n2k2

0Ex (12.21)

−∇2Ey + ∂

∂y(∇ · E) = n2k2

0Ey (12.22)

where ∇2Ex = ∂2Ex

∂x2 + ∂2Ex

∂y2 + ∂2Ex

∂z2 . We will analyse the behaviour of Ez component. In theabove equations, term ∇ · E is expanded as

∇ · E =∂Ex

∂x+ ∂Ey

∂y+ ∂Ez

∂z


and the last term is replaced using relation (12.18). One finds

∇ · E =∂Ex

∂x+ ∂Ey

∂y− 1

n2

∂

∂xn2Ex − 1

n2

∂

∂yn2Ey (12.23)

Substituting (12.23) into (12.21), one obtains

∂

∂x

1

n2

∂

∂xn2Ex + ∂2Ex

∂y2+ ∂2Ex

∂z2+ ∂

∂x

1

n2

∂

∂yn2Ey − ∂2Ey

∂x∂y+ n2k2

0Ex = 0 (12.24)

Similarly, from (12.22) we have

∂

∂y

1

n2

∂

∂xn2Ex − ∂2Ex

∂y∂x+ ∂2Ey

∂x2+ ∂2Ey

∂z2+ ∂

∂y

1

n2

∂

∂yn2Ey + n2k2

0Ey = 0 (12.25)

The above equations in a matrix form are⎡⎢⎢⎣∂

∂x

1

n2

∂

∂xn2 + ∂2

∂y2+ ∂2

∂z2+ n2k2

0

∂

∂x

1

n2

∂

∂yn2 − ∂2

∂x∂y∂

∂y

1

n2

∂

∂xn2 − ∂2

∂y∂x

∂

∂y

1

n2

∂

∂yn2 + ∂2

∂x2+ ∂2

∂z2+ n2k2

0

⎤⎥⎥⎦[Ex

Ey

]= 0

(12.26)Those equations form the starting point for detailed approximations.

12.2.2 Slowly varying envelope approximation (SVEA)

Assume that the transversal field components Ex and Ey are of the form

Ex(x, y, z) = ux(x, y, z)e− jβz (12.27)

Ey(x, y, z) = uy(x, y, z)e− jβz (12.28)

SVEA requires that ∣∣∣∣∂2ui

∂z2

∣∣∣∣ � 2β

∣∣∣∣∂ui

∂z

∣∣∣∣ (12.29)

Applying SVEA to term ∂2

∂z2 where (i = x, y), one obtains

∂2

∂z2Ei = ∂2

∂z2uie

− jβz = e− jβz

{∂2ui

∂z2− 2 jβ

∂ui

∂z− β2ui

}� e− jβz

{−2 jβ

∂ui

∂z− β2ui

}Using SVEA in Eqs. (12.24) and (12.25) results in the paraxial wave equations [9]

j∂ux

∂z= Axxux + Axyuy (12.30)

j∂uy

∂z= Ayxux + Ayyuy (12.31)

297 General theory

where the differential operators are defined by [9]

Axxux = 1

2β

{∂

∂x

1

n2

∂

∂xn2

x + ∂2x

∂y2+ ∂2

x

∂z2+ (

n2k20 − β2

)}ux (12.32)

Axyuy = 1

2β

{∂

∂x

1

n2

∂

∂yn2

y − ∂2y

∂x∂y

}uy (12.33)

Ayxux = 1

2β

{∂

∂y

1

n2

∂

∂xn2

x − ∂2x

∂y∂x+ ∂2

y

∂x2

}ux (12.34)

Ayyuy = 1

2β

{∂2

y

∂x2+ ∂2

y

∂z2+ ∂

∂y

1

n2

∂

∂yn2

y + (n2k2

0 − β2)}

uy (12.35)

The above can be written in a matrix form:

j∂

∂z

[ux

uy

]=[

Axx Axy

Ayx Ayy

] [ux

uy

](12.36)

The evolution of electric field described by Eq. (12.36) is known as vectorial BPM [9]. Ittakes into account the polarization (Axx �= Ayy) and includes coupling between Ex and Ey

(Axy �= Ayx).

12.2.3 Semi-vector BPM

Often the coupling between two polarizations is weak and may be neglected. The twopolarizations are decoupled as long as they are not synchronized with some mechanismwithin the device. The resulting equations which describe semi-vector formulation are

j∂ux

∂z= Axxux

j∂uy

∂z= Ayyuy

Under the semi-vector approximation, the polarization dependencies of the EM waves aretaken into account.

12.2.4 Scalar formulation

If the structure is weakly guiding and/or the polarization dependence is not important, wecan neglect it. The resulting scalar approximation is described by the equation

j∂u

∂z= Au (12.37)

where the operator A is

A = 1

2β

{∂2

x

∂x2+ ∂2

x

∂y2+ (

n2k20 − β2

)}The above reproduces a simple theory summarized in the introductory section.


We finish this section with the remark that similar equations can be derived for themagnetic field, known as the H-formulation (consult [9] for more details).

12.2.5 Finite-difference (FD) approximations

In this section we provide finite-difference discretization of the above equations. Due to itssimplicity, the FD is the popular numerical method of solving such problems. One replacescontinuous space by a discrete lattice where all fields are determined at the lattice’s points.The lattice points are defined by xi = i · �x and y j = j · �y, see Fig. 12.4. Some detailsof discretization are provided in Appendix 12A. The final discrete form of the previousequations for E-formulation is [9] (here we made a replacement ε = n2)

Axxux

= 1

2β

{Ti, j+1ux(i, j + 1) − [

2 − Ri, j+1 − Ri−1, j]

ux(i, j) + Ti−1, jux(i − 1, j)

�x2

+ ux(i, j + 1) − 2ux(i, j) + ux(i, j − 1)

�y2+ [

εi, j,k − β2]

ux(i, j)

}(12.38)

Ayyuy

= 1

2β

{Ti+1, jux(i + 1, j) − [

2 − Ri+1, j − Ri−1, j

]uy(i, j) + Ti, j−1uy(i, j − 1)

�y2

+ uy(i + 1, j) − 2uy(i, j) + uy(i − 1, j)

�x2+ [

εi, j,k − β2]

k2uy(i, j)

}(12.39)

Axyuy

= 1

8β�x�y

{(εi+1, j+1,k

εi+1, j,k− 1

)ux(i + 1, j + 1) −

(εi+1, j−1,k

εi+1, j,k− 1

)ux(i + 1, j − 1)

−(

εi−1, j+1,k

εi−1, j,k− 1

)ux(i − 1, j + 1) +

(εi−1, j−1,k

εi−1, j,k− 1

)ux(i − 1, j − 1)

}(12.40)

Ayxux

= 1

8β�x�y

{(εi+1, j+1,k

εi, j+1,k− 1

)ux(i + 1, j + 1) −

(εi−1, j+1,k

εi, j−1,k− 1

)ux(i − 1, j + 1)

−(

εi+1, j−1,k

εi, j+1,k− 1

)ux(i + 1, j − 1) +

(εi−1, j−1,k

εi, j−1,k− 1

)ux(i − 1, j − 1)

}(12.41)

In the previous equations

Ti±1, j = 2εi±1, j,k

εi±1, j,k + εi, j,k(12.42)

Ri+1, j = Ti±1, j − 1 (12.43)

299 The 1 + 1 dimensional FD-BPM formulation

are the transmission and reflection coefficients across index interfaces between points i andi + 1. Similarly,

Ti, j±1 = 2εi, j±1,k

εi, j±1,k + εi, j,k(12.44)

Ri, j±1 = Ti, j±1 − 1 (12.45)

are the transmission and reflection coefficients across index interfaces between points j andj + 1.

12.3 The 1 + 1 dimensional FD-BPM formulation

If we can neglect y-dependence of the refractive index (wide waveguides), the scalarformulation described by Eq. (12.37) can be further simplified to the 1 + 1 description.Cross-sectional dependence of the refractive index is n = n(x, z). The relevant Helmholtzequation is [10]

2 jβ∂u

∂z= ∂2

x u

∂x2+ (

n2k20 − β2

)u (12.46)

where u is the only electric field component of the TE mode of the waveguide. This equationwill now be analysed using two methods. The second method will be implemented.

12.3.1 Simple approach

Here we discuss the FD-BPM formulation as described by Chung and Dagli [10]. First,replace continuous field u(z, x) by its discrete values as

ui ≡ u(i · �x, z), i = 0, 1, 2 . . . N − 1

Second derivative is approximated as

∂2u

∂x2= ui−1 − 2ui + ui+1

�x2

The resulting finite difference equation is

2 jβ∂ui

∂z= ui−1 − 2ui + ui+1

�x2+ (

n2i k2

0 − β2)

ui ≡ fi(z)

Integrate the equation using trapezoidal rule

2 jβ

∫ ui(z+�z)

ui(z)dui =

∫ z+�z

zfi(z)dz

and have

2 jβ [ui(z + �z) − ui(z)] = 1

2�z [ fi(z + �z) + fi(z)]


Using the definition of fi(z) in the above formula and combining relevant terms, one finds[10]

−aui+1(z + �z) + bui(z + �z) − aui−1(z + �z) = aui+1(z) + cui(z) + aui−1(z)

(12.47)

where

a = �z

2�x2

b = �z

�x2− 1

2�z[k2

0n2i (z + �z) − β2

]+ 2 jβ

c = − �z

�x2+ 1

2�z[k2

0n2i (z) − β2

]+ 2 jβ

The scheme results in a tridiagonal system of linear equations. For more details, see [10].

12.3.2 Propagator approach

Approach 1

When one approximates propagation within the waveguide by a one-dimensional approachusing variables (z, x) with z as the propagation direction, the y dependence is irrelevant andone then sets ∂/∂y = 0. The resulting equation is

2 jβ∂u

∂z= ∂2u

∂x2+ (k2 − β2)u (12.48)

where k = nk0.The finite-difference discretization of Eq. (12.48) in 1D is the following:

jun+1

i − uni

h= 1

2β

uni+1 − 2un

i + uni−1

�x2+ 1

2β(k2

i − β2) (12.49)

where k2i = k2

0 n2i (x) ≡ k2

0 n2(xi). With an introduction of the propagation operator P, theabove scheme is

jun+1

i − uni

h=

N∑k=1

Pik unk (12.50)

The matrix elements of the operator P are

Pik = 1

2β

1

�x2(δi+1,k − 2 δi,k + δi−1,k ) + 1

2β(k2

i δi,k − β2) (12.51)

The solution of Eq. (12.50) can be written as

−→u n+1 =(←→

I − jh←→P)−→u n (12.52)

where −→u n is a column vector which consists of elements uni ,

←→I is the identity matrix and←→

P is the propagation matrix operator having elements given by Eq. (12.51). The schemeis known as the explicit scheme. It is numerically unstable if the h step is too large.


The improvement is made by applying operator P to the future value of u [11]. FromEq. (12.50)

jun+1

i − uni

h=

N∑k=1

Pik un+1k (12.53)

or

−→u n+1 = −→u n − jh←→P −→u n+1

or (←→I + jh

←→P)−→u n+1 = −→u n

The solution is

−→u n+1 =(←→

I + jh←→P)−1 −→u n (12.54)

The scheme is called implicit method, and it is stable, see Problem.The combination of the above two methods is created by taking the average between

implicit and explicit schemes. The new scheme is known as the Crank-Nicolson method[11]. It is both more accurate and stable. Adding Eqs. (12.50) and (12.53), one obtains (theCrank-Nicolson scheme)

2 jun+1

i − uni

h=

N∑k=1

Pik

(un

k + un+1k

)(12.55)

In matrix form

−→u n+1 = −→u n − 1

2jh

←→P(un

k + un+1k

)(12.56)

The above can be expressed as(←→I + 1

2jh

←→P

)−→u n+1 =

(←→I − 1

2jh

←→P

)−→u n (12.57)

or

L+ −→u n+1 = L− −→u n (12.58)

where

L+ = ←→I + 1

2jh

←→P (12.59)

and

L− = ←→I − 1

2jh

←→P (12.60)

From that equation, the final form which can be implemented is

−→u n+1 = L−1+ L− −→u n (12.61)


Approach 2

The above derivation can be made more formal. For that purpose, write Eq. (12.48) as

j∂u

∂z= P · u (12.62)

where the propagation operator P is defined as

P = 1

2β

[∂2

x + (k2 − β2)]

(12.63)

The solution of Eq. (12.62) is

u(z) = e− jzPu(0)

or, when one introduces ‘small’ step h:

u(z + h) = e− jhPu(z) (12.64)

For a small step, the solution (12.64) can be approximated as

u(z + h) ≈ (1 − jhP) u(z) (12.65)

One might notice that the solution (12.64) can also be written as

e jhPu(z + h) = u(z) (12.66)

or

(1 + jhP) u(z + h) ≈ u(z) (12.67)

or

u(z + h) ≈ (1 + jhP)−1 u(z) (12.68)

To solve this problem, a practically important method is known as the Crank-Nicolsonscheme which is obtained from the solutions (12.65) and (12.68). One thus makes thefollowing manipulations:

u(z + h) =(

1 + jh

2P

)−1

u

(z + h

2

)=(

1 + jh

2P

)−1 (1 + jh

2P

)u(z) (12.69)

using Eq. (12.65) for h → h/2. The implementation of this scheme is shown in Appendix,Listing 12A.2. It is used to propagate a Gaussian pulse in an empty space. In our im-plementation we assumed that β = n · k0. With this choice, last term in propagator Pvanishes.

Profiles of a Gaussian pulse at various locations along the propagation direction in afree space are shown in Fig. 12.7. One can observe reduction of a peak value and also theeffects of reflections from the boundaries of the computational window. Elimination of thosereflections will be discussed in the next sections. In Fig. 12.8 we show a three-dimensionalview of profiles of Gaussian pulses shown in Fig. 12.7.


0 50 100 1500

0.2

0.4

0.6

0.8

1

Fig. 12.7 Profiles of Gaussian pulses at various locations along z-axis.

−5

0

5

0

20

400

0.5

1

x (mm)z (mm)

Fig. 12.8 Three-dimensional view of profiles shown in Fig. 12.7 of Gaussian pulses propagating in a free space.

Approach 3

Before starting the discussion of boundary conditions, we want to summarize yet anotherpossible numerical approach to Eq. (12.48). It will then be used to derive details of trans-parent boundary conditions.

Assuming an equidistant step along x-axis and using standard formulas (h ≡ �z)

∂u

∂z= un+1

i − uni

h(12.70)

∂2u

∂x2= ui+1 − 2ui + ui−1

�x2(12.71)


the discretized version of Eq. (12.48) is

2 jβun+1

i − uni

h= ui+1 + ui−1

�x2+[− 2

�x2+ (

k2i − β2

)]ui (12.72)

Introduce modifications on the right hand side of Eq. (12.72):

ui+1 = 1

2

(un+1

i+1 + uni+1

)(12.73)

ui−1 = 1

2

(un+1

i−1 + uni−1

)(12.74)

k2i = 1

2

[k2

i (n + 1) + k2i (n)

](12.75)

After new algebraic steps, Eq. (12.72) can be expressed as

un+1i + jh

2

{1

2β

1

�x2

(un+1

i+1 − 2un+1i + un+1

i−1

)+ 1

2β

[k2

i (n + 1) − β2]

un+1i

}= un

i − jh

2

{1

2β

1

�x2

(un

i+1 − 2uni + un

i−1

)+ 1

2β

[k2

i (n) − β2]

uni

}(12.76)

Note the different n dependence in k2i . The final form of the above in a condensed form is

L+(n + 1)−→u n+1 = L−(n)

−→u n (12.77)

where

L+(n + 1)−→u n+1 = un+1

i + jh

2

{1

2β

1

�x2

(un+1

i+1 − 2un+1i + un+1

i−1

)+ 1

2β

[k2

i (n + 1) − β2]

un+1i

}(12.78)

and

L−(n)−→u n = un

i − jh

2

{1

2β

1

�x2

(un

i+1 − 2uni + un

i−1

)+ 1

2β

[k2

i (n) − β2]

uni

}(12.79)

12.3.3 Transparent boundary conditions

To eliminate reflections shown in the previous example, one needs to introduce appropriateboundary conditions outside the computational window. The popular solution is to introducethe so-called transparent boundary conditions (TBC) which were invented by Hadley [12],[13]. TBC have been extensively discussed in the literature [14], [15], [16], [17].

Consider the system with left and right boundaries as shown in Fig. 12.9. Nodes x0 andxN+1 are outside the system; however, they are needed in the implementation of numericalscheme.


i=0 1 2 3 N N+1N-1

x0

xx1 x3 xN-1 xN xN+1x2

Fig. 12.9 Numbering of nodes used in deriving transparent boundary conditions.

Left-hand boundary

Analyse the left boundary first. Assume that near the boundary the field is approximated as

u(x, z) ≈ A(z) eikxx (12.80)

For the first left points shown in Fig. 12.9, one obtains

u0 = u(x0) = A(z) eikxx0 (12.81)

u1 = u(x1) = A(z) eikxx1 (12.82)

u2 = u(x2) = A(z) eikxx2 (12.83)

From the above equations one finds

u2

u1= eikxx2

eikxx1= eikx(x2−x1) ≡ eikx�x (12.84)

andu1

u0= eikxx1

eikxx0= eikx(x1−x0) ≡ eikx�x (12.85)

assuming uniform grid; i.e. �x = x2 − x1 = x1 − x0. From the above equations, onedetermines value of the field at the outside point x0:

u0 = u1 eikx�x (12.86)

Wave number kx is determined from known fields using Eq. (12.84):

kx = 1

i�xln

u2

u1(12.87)

For a field travelling leftward (an outgoing wave), the real part of kx, Re(kx) is positive.The implementation of TBC at the left point for operator Ln+1

+ will be illustrated now.Consider the first element for i = 1, which from Eq. (12.78) is

Ln+1+ (1) = un+1

1 + 1

2jh

1

2β

1

�x2(−2)un+1

1 + 1

2jh

1

2β

1

�x2un+1

0 (12.88)

Replacing un+10 using Eq. (12.86) gives

Ln+1+ (1) = un+1

1 + 1

2jh

1

2β

1

�x2(−2)un+1

1 + 1

2jh

1

2β

1

�x2un+1

1 e jkx�x

=(

1 + 1

2jh

1

2β

1

�x2(−2) + 1

2jh

1

2β

1

�x2e jkx�x

)un+1

1

= (old term + left correction) un+11 (12.89)


The last expression is directly implemented. For operator Ln+1− there is only a change of

sign. Similar steps can be taken for the right boundary, which is left as a problem.

Right-hand boundary

At the right-hand boundary, the point xN+1 is outside the system and the field there mustbe determined. One starts with assuming the following expression for the right-travellingfield:

u(x) ≈ A(z) e−ikxx (12.90)

Based on this assumption, we can write fields at three points around the right boundary:

uN−1 = u(xN−1) = A(z) e−ikxxN−1 (12.91)

uN = u(xN ) = A(z) e−ikxxN (12.92)

uN+1 = u(xN+1) = A(z) e−ikxxN+1 (12.93)

Here uN−1 and uN are fields within the system and uN+1 is outside. From the above relationsone determines the outside field as

uN+1 = uN e−ikx�x (12.94)

where the wavenumber is

kx = 1

i�xln

u2

u1(12.95)

Cross sections are shown in Fig. 12.10. One can observe that reflections were eliminated.The three-dimensional view of an initial Gaussian pulse at various positions is shown inFig. 12.11. MATLAB code is shown in Appendix, Listings 12B.3 and 12B.3.1.

12.4 Concluding remarks

A few years ago an article was published with the title ‘What is the future for beampropagation methods?’ [18]. The role played by BPM in the simulations of photonicdevices was discussed. The authors also provided an extensive literature summary on thefollowing issues associated with the BPM: transverse discretization, explicit and implicitformulations, vector effects, wide angle effects, reflective schemes and time domain BPM.They concluded with highlighting the outstanding issues (as of 2004) of BPM, whichwere: flexible meshing, development of wide-angle schemes, improvement of boundaryconditions and creation of hybrid BPM schemes, e.g. with time and/or frequency domainmethods.

A more recent assessment of BPM (and also Eigenmode Expansion Method and FiniteDifference Time Domain) for Photonic CAD is reported by Gallagher [19]. He summa-rizes the main algorithms used in photonic modelling and discusses their strengths andweaknesses.

307 Problems

0 50 100 1500

0.2

0.4

0.6

0.8

1

Fig. 12.10 Comparison of field profiles at various locations along z-axis of Gaussian pulse propagating in a free space by BPMwithtransparent boundary conditions. One observes reduction of its peak value and spreading of pulse. No reflections areobserved.

−5

0

5

0

20

400

0.5

1

xz

Fig. 12.11 Three-dimensional view of Gaussian pulse shown in Fig. 12.10 propagating in a free space by BPMwith transparentboundary conditions.

12.5 Problems

1. Develop H-formulation of BPM.2. Perform von Neumann stability analysis for the implicit method.3. Based on the example in the text for TBC for the left boundary, develop expressions

for the right TBC.


12.6 Project

1. Implement 1-1 FD-BPM as discussed in Section 12.3.1. Consult work by [10]. Analysewaveguides described in Ref. [10].

Appendix 12A: Details of derivation of the FD-BPM equation

We describe details of discretization of the earlier equations, like Eq. (12.32). We presentdetails for the 1D case only. Generalization to 2D is straightforward. A typical term whichneeds discretization is

K = ∂

∂x

1

ε

∂

∂x(εu) ≡ ∂

∂xg(x)

Simple approach

First-order derivative is∂φ

∂x= φp+1 − φp

�x

where φ = εu. Applying the above formula with difference centred around the points p+ 12

and p − 12 , one finds

K = 1

�x

{1

εp+ 12

φp+1 − φp

�x− 1

εp− 12

φp − φp−1

�x

}where

φp±1 = εp±1up±1

φp = εpup

We also use the following intuitive relations (which will be proved in the next section):

εp+ 12

= εp+1 + εp

2

εp− 12

= εp+1 + εp−1

2

Combining the above relations, one obtains

K = 1

�x

2

εp+1 + εp

εp+1up+1 − εpup

�x− 1

�x

2

εp + εp−1

εpup − εp−1up−1

�x

= 2

�x2

{εp+1

εp+1 + εpup+1 −

[εp

εp+1 + εp+ εp

εp + εp−1

]up + εp−1

εp + εp−1up−1

}

309 Appendix 12A

Taylor series approach

We differentiate

K = ∂

∂x

1

ε

∂

∂x(εu) = ∂

∂x

1

ε

(∂ε

∂x

)u + ∂2u

∂x2

First term is evaluated as follows. First, we expand around point p:(1

ε

(∂ε

∂x

)u

)p± 1

2

=(

1

ε

(∂ε

∂x

)u

)p

± �x

2

∂

∂x

1

ε

(∂ε

∂x

)u

∣∣∣∣p

Subtracting gives

∂

∂x

1

ε

(∂ε

∂x

)u

∣∣∣∣p

= 1

�x

{(1

ε

(∂ε

∂x

)u

)p+ 1

2

−(

1

ε

(∂ε

∂x

)u

)p− 1

2

}(12.96)

From the above, one can observe that the values of u and ε are needed at intermediate pointsp + 1

2 and p − 12 . Those can be obtained by performing expansions

up+1 = up+ 12+ ∂u

∂x

∣∣∣∣p+ 1

2

�x

2

and

up = up+ 12− ∂u

∂x

∣∣∣∣p+ 1

2

�x

2

Adding the above gives

up+ 12

= 1

2(up + up+1) (12.97)

Expansions of ε gives

εp+1 = εp+ 12− ∂ε

∂x

∣∣∣∣p+ 1

2

�x

2

and

εp = εp+ 12− ∂ε

∂x

∣∣∣∣p+ 1

2

�x

2

Adding the above gives

εp+ 12

= 1

2(εp + εp+1) (12.98)

Subtracting gives

∂ε

∂x

∣∣∣∣p+ 1

2

= 1

2(εp+1 − εp) (12.99)


From Eqs. (12.98) and (12.99) we have

1

ε

∂ε

∂x

∣∣∣∣p+ 1

2

= 2

�x

εp+1 − εp

εp + εp+1(12.100)

Using Eqs. (12.97) and (12.100), we obtain

1

ε

∂ε

∂xu

∣∣∣∣p+ 1

2

= 1

�x

εp+1 − εp

εp + εp+1(up + up+1) (12.101)

In exactly the same way, performing the relevant expansions around point p − 12 one

finds1

ε

∂ε

∂xu

∣∣∣∣p− 1

2

= 1

�x

εp − εp−1

εp + εp−1(up + up−1) (12.102)

Substituting Eqs. (12.101) and (12.102) into (12.96), we obtain an intermediate result

∂

∂x

1

ε

(∂ε

∂x

)u

∣∣∣∣p

= 1

�x2

{εp+1 − εp

εp + εp+1(up + up+1) − εp − εp−1

εp + εp−1(up + up−1)

}(12.103)

A complete derivative is obtained by combining discretization for a second derivative

∂2u

∂x2= 1

�x2

(up+1 − 2up + up−1

)with expression (12.103). The result is

K = ∂

∂x

1

ε

∂

∂x(εu)

= 2

�x2

{εp+1

εp + εp+1up+1 − εp

(1

εp + εp+1+ 1

εp + εp+1

)up + εp−1

εp + εp−1up−1

}The last result is the same as the one obtained by Lidgate [20].

Appendix 12B: MATLAB listings




12B.1 pbpm.m Gaussian pulse in free space in the paraxial approximation12B.2 f d bpm f ree.m Gaussian pulse in free space (Crank-Nicholson)12B.3 bpm tbc.m Transparent boundary conditions12B.3.1 prop.m Function called by bpm tbc.m. Performs single step

311 Appendix 12B

Listing 12B.1 Function pbpm.m. Function performs propagation of a two-dimensionalGaussian pulse using the Fourier transform split-step beam propagation method in theparaxial approximation.

% File name: pbpm.m

% Propagation of 2D Gaussian pulse by paraxial FT split-step BPM

clear all N= 10; delta_x = 1/N;

x = -5:delta_x:5; % creation of space arguments

y = -5:delta_x:5; % creation of space arguments

delta_z = 5.0; % step size along z-axis

delta_n = 0.4; k_zero = 100;

beta = 20; % propagation constant

%

u_init=(exp(-x.^2))’*exp(-y.^2); % initial Gaussian pulse

mesh(x,y,abs(u_init)); % plots original pulse

pause close all

%

k_x = -5:1/N:5; % creation of Fourier variables

k_y = -5:1/N:5; % creation of Fourier variables

%

temp = delta_z/(2*beta); H_transfer =

(exp(1i*temp*k_x.^2))’*(exp(1i*temp*k_y.^2)); H_transfer =

fftshift(H_transfer); S_phase = exp(-1i*k_zero*delta_n);

%

for n=1:100

z = fft2(u_init);

zz = z.*H_transfer;

u_prime = ifft2(zz);

u = S_phase.*u_prime;

u_init = u;

end

%

mesh(x,y,abs(u_init)) % plots pulse after propagation

pause close all

Listing 12B.2 Program fd bpm free.m. Illustrates propagation of a Gaussian pulse in afree space.

% File name: fd_bpm_free.m

% Propagation of Gaussian pulse in a free space by Crank-Nicholson method

% No boundary conditions are introduced

clear all

L_x=10.0; % transversal dimension (along x-axis)

w_0=1.0; % width of input Gaussian pulse

lambda = 0.6; % wavelength


n=1.0; % refractive index of the medium

k_0=2*pi/lambda; % wavenumber

N_x=128; % points on x axis

Delta_x=L_x/(N_x-1); % x axis spacing

h=5*Delta_x; % propagation step along z-axis

N_z=100; % number of propagation steps

plotting=zeros(N_x,N_z); % storage for plotting

x=linspace(-0.5*L_x,0.5*L_x,N_x); % coordinates along x-axis

x = x’;

E=exp(-(x/w_0).^2); % initial Gaussian field

%

% beta = n*k_0. With this choice, last term in propagator vanishes

prefactor = 1/(2*n*k_0*Delta_x^2); main = ones(N_x,1); above =

ones(N_x-1,1); below = above;

P = prefactor*(diag(above,-1)-2*diag(main,0)+diag(below,1)); % matrix P

%

step_plus = eye(N_x) + 0.5i*h*P; % step forward

step_minus =eye(N_x)-0.5i*h*P; % step backward

%

z = 0; z_plot = zeros(N_z); for r=1:N_z

z = z + h;

z_plot(r) = z + h;

plotting(:,r)=abs(E).^2;

E=step_plus\step_minus*E;

end;

%

for k = 1:N_z/10:N_z % choosing 2D plots every 10-th step

plot(plotting(:,k),’LineWidth’,1.5)


hold on

end pause close all

%


y = z_plot(k)*ones(size(x)); % spread out along y-axis

plot3(x,y,plotting(:,k),’LineWidth’,1.5)

hold on

end grid on xlabel(’x (mm)’,’FontSize’,14)

ylabel(’z (mm)’,’FontSize’,14) % along propagation direction


pause close all

Listing 12B.3 Program bpm tbc.m. Illustrates propagation of Gaussian pulse in a freespace with transparent boundary conditions.

% File name: bpm_tbc.m

% Illustrates propagation of Gaussian pulse in a free space

313 Appendix 12B

% using BPM with transparent boundary conditions

% Operator P is determined in a separate function

clear all




n=1.0; % refractive index of the medium


N_x=128; % number of points on x axis


h=5*Delta_x; % propagation step




x = x’;

E=exp(-(x/w_0).^2); % initial Gaussian field

%

z = 0;

z_plot = zeros(N_z);

for r=1:N_z % BPM stepping

z = z + h;

z_plot(r) = z + h;

plotting(:,r)=abs(E).^2;

E = step(Delta_x,k_0,h,n,E); % Propagates pulse over one step

end;

%


plot(plotting(:,k),’LineWidth’,1.5)


hold on

end

pause

close all

%



plot3(x,y,plotting(:,k),’LineWidth’,1.5)

hold on

end

grid on

xlabel(’x’,’FontSize’,14)

ylabel(’z’,’FontSize’,14) % along propagation direction


pause

close all


Listing 12B.3.1 Function step.m used by bpm tbc.m. Function performs a single step inBPM.

% File name: step.m

function E_new = step(Delta_x,k_0,h,n,E_old)

% Function propagates BPM solution along one step

N_x = size(E_old,1); % determine size of the system

%--- Defines operator P outside of a boundary

prefactor = 1/(2*n*k_0*Delta_x^2);

main = ones(N_x,1);

above = ones(N_x-1,1);

below = above;


%

L_plus = eye(N_x) + 0.5i*h*P; % step forward

L_minus = eye(N_x)-0.5i*h*P; % step backward

%

%---- Implementation of boundary conditions

%

pref = 0.5i*h/(2*k_0*Delta_x^2);

k=1i/Delta_x*log(E_old(2)/E_old(1));

if real(k)<0

k=1i*imag(k);

end;

left = pref*exp(1i*k*Delta_x); % left correction for next step

L_plus(1) = L_plus(1)+left;

L_minus(1) = L_minus(1)-left;

%

k=-1i/Delta_x*log(E_old(N_x)/E_old(N_x-1));

if real(k)<0

k=1i*imag(k);

end;

right = pref*exp(1i*k*Delta_x); % right correction for nest step

L_plus(N_x) = L_plus(N_x) + right;

L_minus(N_x) = L_minus(N_x) - right;

%

E_new = L_minus\L_plus*E_old; % determine new solution

References

[1] M. D. Feit and J. A. Fleck Jr. Light propagation in grated0index optical fibres. Appl.Opt., 17:3990–8, 1978.

315 References

[2] V. P. Tzolov, D. Feng, S. Tanev, and Z. J. Jakubczyk. Modeling tools for integratedand fiber optical devices. In G. C. Righini and S. Iraj Najafi, eds., Integrated OpticsDevices III. Proceedings of SPIE, volume 3620, pages 162–73. EMW Publishing,Cambridge, MA, 1999.

[3] K. Kawano and T. Kitoh. Introduction to Optical Waveguide Analysis. SolvingMaxwell’s Equations and the Schroedinger Equation. Wiley, New York, 2001.


[5] K. Okamoto. Fundamentals of Optical Waveguides. Academic Press, Amsterdam,2006.

[6] G. Lifante. Integrated Photonics. Fundamentals. Wiley, Chichester, 2003.[7] R. L. Liboff. Introductory Quantum Mechanics. Addison-Wesley, Reading, MA, 1992.[8] C. L. Xu and W. P. Huang. Finite-difference beam propagation method for guide-

wave optics. In J. A. Kong, ed., Progress in Electromagnetic Research, PIER 11,pages 1–49. EMW Publishing, Cambridge, MA, 1995.

[9] W. P. Huang and C. L. Xu. Simulation of three-dimensional optical waveguides by afull-vector beam propagation method. IEEE J. Quantum Electron., 29:2639–49, 1993.

[10] Y. Chung and N. Dagli. An assessment of finite difference beam propagation method.IEEE J. Quantum Electron., 26:1335–9, 1990.

[11] A. L. Garcia. Numerical Methods for Physics. Second Edition. Prentice Hall, UpperSaddle River, 2000.

[12] G. R. Hadley. Transparent boundary condition for beam propagation. Opt. Lett.,16:624–6, 1991.

[13] G. R. Hadley. Transparent boundary condition for the beam propagation method.IEEE J. Quantum Electron., 28:363–70, 1992.

[14] F. Schmidt and P. Deuflhard. Discrete transparent boundary conditions for the numer-ical solution of Fresnel’s equation. Computers Math. Applic., 29:53–76, 1995.

[15] F. Fogli, G. Bellanca, and P. Bassi. TBC and PML conditions for 2D and 3D BPM: acomparison. Optical and Quantum Electronics, 30:443–56, 1998.

[16] D. Yevick, T. Friese, and F. Schmidt. A comparison of transparent boundary conditionsfor the Fresnel equation. Journal of Computational Physics, 168:433–44, 2001.

[17] F. Schmidt, T. Friese, and D. Yevick. Transparent boundary conditions for split-stepPade approximations of the one-way Helmholtz equation. Journal of ComputationalPhysics, 170:696–719, 2001.

[18] T. M. Benson, B. B. Hu, and P. Sewell. What is the future for beam propagation meth-ods? In G. Lampropoulos, J. Armitage, and R. Lessard, eds., Proc. of SPIE PhotonicsNorth 2004: Photonic Applications in Telecommunications, Sensors, Software andLasers, volume 5579, pages 351–8. SPIE, Bellingham, WA, 2004.

[19] D. Gallagher. Photonic CAD matures. IEEE LEOS Newsletter, 2:8–14, 2008.[20] S. Lidgate. Advanced Finite Difference – Beam Propagation Method Analysis of

Complex Components. PhD thesis, Nottingham, 2004.

13 Somewavelength divisionmultiplexing(WDM) devices

Wavelength division multiplexing (WDM) is a modern practical method of increasingtransmission capacity in fibre communication systems. It uses the principle that opticalbeams with different wavelengths can propagate simultaneously over a single fibre withoutinterfering with one another. In the wavelength range of 1280–1650 nm (like an AllWavefibre [1]) the useable bandwidth of a single mode fibre is about 53 THz. In recent years animproved (denser) WDM system known as DWDM is under development.

In this chapter we discuss some of the WDM devices and also provide some applicationsof BPM developed earlier to simulate those devices. We start by summarizing the basicWDM system.

13.1 Basics of WDM systems

WDM is the main technique used in the realization of all optical networks. WDM is thetechnology which combines a number of wavelengths onto the same fibre.

Key features include:

• capacity upgrade• transparency (each optical channel can carry any transmission format)• wavelength routing• wavelength switching

Implementation of a typical WDM system employing N channels is shown in Fig. 13.1.In the shown system three wavelengths are multiplexed in one fibre to increase transmissioncapacity. The light of laser diodes with wavelengths recommended by the ITU is launchedinto the inputs of a wavelength multiplexer (MUX) where all wavelengths are combinedand coupled into a single-mode fibre. When needed, propagating light can be amplified byan optical fibre amplifier and eventually imputed at the wavelength demultiplexer (DMUX)which separates all optical channels and sends them to different outputs.

In order to find the optical bandwidth corresponding to a spectral width in optical region,we use the relation c = λ · ν, where λ is wavelength and ν carrier frequency and c velocityof light. Differentiating

dν = cd

dλ

(1

λ

)dλ = − c

λ2dλ

316

317 Basic WDM technologies

RX

RX

RX

TX

TX

TX

Optical fibreMultiplexer

λ1 λ1

λ2

λ3

Demultiplexer

Fig. 13.1 Implementation of a typical WDM link.

or

|�ν| = c

λ2|�λ| (13.1)

The above equation describes the frequency change �ν which corresponds to the wave-length change �λ around λ. Using the above formula, we can estimate the usable wavelengthrange for a standard single-mode fibre. Assuming that telecommunication wavelength rangeextends from λ1 = 1280 nm to λ2 = 1625 nm, the ultimate bandwidth of optical fibre is40 THz [2]. Assuming 50 or 25 GHz channel spacing, there is the possibility to transmit800–1600 wavelength channels.

In the remainder of this chapter, we will consider some of the basic devices used in WDMsystems and provide two applications of the beam propagation method (BPM) to simulatesimple structures.

13.2 Basic WDM technologies

In an ideal WDM system where nonlinear effects are neglected, the discrete wavelengthscan be optically processed, i.e. routed and/or switched without interfering with each other.Optical processing can be performed using passive or active components.

One refers to passive components where there is no external control of their operation.Typically, they are used to split or combine signals. Examples of passive components are:N × N couplers, power splitters and star couplers.

Active components, which can be controlled electronically, include tunable optical filters,tunable sources, optical amplifiers. Here, we will concentrate on passive components.

Basic technologies which are used to develop WDM devices are [3]: fibre Bragg grating(FBG), array waveguide grating (AWG), thin-film filter (TFF) and diffraction grating (DG).

Another classification of main WDM components used in fibre optic communicationsystems is described by Agrawal [4] and includes: tunable optical filters (Fabry-Perotfilters, Mach-Zehnder filters, grating-based filters, application-based filters), multiplexersand demultiplexers, add/drop multiplexers and filters, broadcast star couplers, wavelengthrouters, optical cross-connects, wavelength converters and WDM transmitters and receivers.

In the following, we will discuss basic WDM technologies. We start with fibre Bragggrating.

318 Some wavelength division multiplexing (WDM) devices

Gratingperiod

λ1 λ2

λB

λ3 Fibrecore

Fig. 13.2 Fibre grating operating as an optical filter.

λ1

λ2

λ3

λ1 λ2 λ3

Fig. 13.3 Schematic illustration of a simple array waveguide demultiplexer.

13.2.1 Fibre Bragg grating

Fibre Bragg grating (FBG) devices are based on the principle discovered by Bragg in 1913and demonstrated in fibre by Hill in 1978 [5]. It is illustrated in Fig. 13.2.

In a fibre’s core a periodic change of the refractive index (known as grating) is createdusing the effect of photosensitivity in germanium-doped optical fibre [5]. When a lightsignal consisting of several wavelengths travels in optical fibre (here from left to right),the signal with a wavelength which obeys the Bragg condition is reflected. The reflectedwavelength centred at λB that fulfills the Bragg condition is

λB = 2�ne f f (13.2)

where � is the grating period and ne f f is the effective group refractive index of the core.Applications of Bragg grating are numerous and include filters and dispersion compen-

sation. A comprehensive review on FBG including fibre grating lasers and amplifiers waswritten by Kashyap [6].

13.2.2 Array waveguide grating

Array waveguide grating (AWG) is shown schematically in Fig. 13.3 [2]. It is formed byseveral waveguides with different lengths. At both ends all waveguides are converging tothe same points. Light composed of several wavelenths (λ1, λ2, . . .) enters the device onthe left. The length of each waveguide is carefully designed so to provide precise phasedifference between neighbouring waveguides at the end of the guiding structure (on theright). At the output, a diffraction pattern is created which allows output waveguide tocollect light at a particular wavelength. AWG can be used as a passive optical multiplexerand/or demultiplexer.


Pin

Pin0.5

Pin0.5

Fig. 13.4 Y-branch power splitter.

Pin0.25

Pin0.25

Pin0.25

Pin0.25

Pin

Fig. 13.5 A four-port splitter.

1 2

34

InputOutput

Output

Fig. 13.6 Integrated optics directional coupler.

13.2.3 Couplers and splitters

Optical couplers are passive devices which either split optical signal into multiple paths, orcombine several signals into one path. A prime characteristic of couplers is the number ofinput and output ports, which is typically expressed as an N × M configuration, where Nrepresents the number of inputs and M represents the number of outputs.

A splitter is a modification of a coupler. As an example, a Y-branch power splitter is shownin Fig. 13.4. Typically, such Y-branch splits power evenly between the two output ports. Bycombining several Y-branches more outputs can be provided, as shown in Fig. 13.5.

Couplers can be constructed within the waveguide structure as shown in Fig. 13.6. Sucha structure can be formed by ion exchange, ion implantation or chemical vapour deposition.

Couplers can also be constructed by twisting two or more fibres together and meltingthem in a flame. The process creates a fused coupler, a very popular device.

These techniques can be used to make 50−50% or 99−1% couplers. The numbersindicate splitting ratios. The length of the coupling region (fused region) as well as thetwisting determines the splitting ratio. Such couplers are simple to construct but fabricationrequires significant experience.

In the coupler shown in Fig. 13.6, the optical signal enters port 1. Some of its power exitsin port 2 and some in port 3. In an ideal situation, no light reaches port 4 and also no poweris lost. In practice, however, a few tenths of a dB are lost and the coupling to port 4 is about30 dB relative to input power at port 1. The percentage of light coupled to different portscan be varied by changing the coupling length L.


a b

zx

y

Fig. 13.7 Two coupled optical waveguides. Distribution of electric fields is also shown.

Some of the common applications of couplers and splitters are

• to monitor output of light locally (usually 99−1% couplers are used)• to distribute an incoming signal to several locations simultaneously. For example, a

four-part splitter (Fig. 13.5) allows a signal to drive four receivers.

Next, we provide a summary of the mathematical description of a passive coupler.

13.2.4 Mathematical theory of a passive coupler

Consider two waveguides ‘a’ and ‘b’ (Fig. 13.7), see [7]. For a single waveguide, say ‘a’,one can express the field as

�E (x, y, z) = �E (a) (x, y) a (z)

�H (x, y, z) = �H (a) (x, y) a (z)

and where

a (z) = a0eiβaz

where �E (a) (x, y) and �H (a) (x, y) are modal distributions in (x, y) plane. They are normalizedas

1

2Re

∫ ∫�E (a)∗ (x, y) × �H (a)∗dxdy · z = 1

Alsoda (z)

dz= iβaa (z)

Total guided power

P = 1

2

∫ ∫�E (a)∗ (x, y, z) × �H ∗ (x, y, z) · zdxdy

= |a (z)|2

For two parallel waveguides, fields in each one are written as

�E (x, y, z) = a (z) �E (a) (x, y) + b (z) �E (b) (x, y) (13.3)

�H (x, y, z) = a (z) �H (a) (x, y) + b (z) �H (b) (x, y) (13.4)


Amplitudes a (z) and b (z) satisfy the following (coupled-mode) equations:

da(z)

dz= iβaa(z) + iκabb(z) (13.5)

db(z)

dz= iκbaa(z) + iβbb(z) (13.6)

where κab and κba are coupling coefficients. Guided power is

P = sa |a (z)|2 + sb | b (z)|2 + Re{a (z) b∗ (z)Cba + b (z) a∗ (z)Cab

}(13.7)

where

Cpq = 1

2

∫ +∞∫−∞

�E (q) (x, y) × �H (p)∗ ( x, y) · zdxdy (13.8)

In the above, sa, sb = +1 is for propagation in the +z direction and sa, sb = −1 is forpropagation in the −z direction.

Coupled mode equations can be written in a matrix form as

d

dz

[a(z)b(z)

]= i

←→M

[a(z)b(z)

](13.9)

where

←→M =

[βa κab

κba βb

](13.10)

The solution is assumed to be [a(z)b(z)

]=[

AB

]eiβz (13.11)

After substitution into Eq. (13.9), one finds[←→M −β

←→1] [A

B

]= 0 (13.12)

where←→1 is an identity matrix. In the full form[

βa − β κab

κba βb − β

] [AB

]= 0 (13.13)

For non-trivial solutions, a determinant must vanish:

det = (βa − β) (βb − β) − κab · κba = 0 (13.14)

From the above, two eigenvalues are found as

β = 1

2(βa + βb) ± γ ≡

{β+β−

(13.15)

where

γ =√

�2 + κab · κba, � = 1

2(βb − βa) (13.16)


Eigenvectors are

�V1 =[

κab

� + γ

]or

[−� + γ

κba

]for β+ (13.17)

and

�V2 =[

κab

� − γ

]or

[−� − γ

κba

]for β− (13.18)

The general solution is therefore[a(z)b(z)

]= ←→

V

[eiβ+z 0

0 eiβ−z

]←→V −1

[a(0)

b(0)

](13.19)

where matrix←→V is formed from eigenvectors as

←→V =

[�V1; �V2

](13.20)

After some algebra, one finds the final solution[a(z)b(z)

]= ←→

S (z)

[a(0)

b(0)

](13.21)

with

←→S (z) =

[cos γ z − i�

γsin γ z i κba

γsin γ z

i κba

γsin γ z cos γ z + i�

γsin γ z

]· e

i2 (βa+βb)z (13.22)

As a special case, consider a situation when at z = 0 the optical power is incident onlyin waveguide 1, a (0) = 1, b (0) = 0. One finds in this case

|b (z)|2 =∣∣∣∣κba

γ

∣∣∣∣2 sin2 γ z

At γ z = π2 , 3π

2 , . . . , (2n + 1) π2 , the power transfer from guide ‘a’ to guide ‘b’ is maximum.

Since ∣∣∣∣κba

γ

∣∣∣∣2 = |κba|2[12 (βb − βa)

]2 + |κba|2< 1

for βa �= βb the power transfer between waveguides is never complete. For more discussionconsult Chuang [7].

13.2.5 Optical isolators

An optical isolator is a passive device which allows the transmission of an optical signalin only one direction. At the same time the reflections in the opposite direction will beeliminated.

Two key parameters of an isolator are: the insertion loss which is the loss in the forwarddirection, and its isolation which is the loss in the reverse direction. Typical insertion lossis about 1 dB, whereas the isolation loss is 40–50 dB.

323 Applications of BPM to photonic devices

Polarizer

Incidentlight

Outputlight

Faraday rotator

Polarizer

Fig. 13.8 Illustration of the principle of Faraday rotation.

An optical isolator based on the Faraday effect is shown in Fig. 13.8. The device consistsof two linear polarizers and a 45◦ Faraday rotator. A light beam entering the device from theleft passes through a linear polarizer which polarizes it vertically. Then it passes through aFaraday rotator which rotates it (here, say by 45◦) with respect to vertical direction. Afterpassing through a rotator, it travels through another linear polarizer aligned at 45◦ whichallows the beam to pass through. Therefore, a vertically polarized beam can be transmittedthrough the device. However, all beams polarized at other angles will be blocked.

The angle of rotation α in a Faraday rotator is given by

α = V BL (13.23)

where V is the Verdet constant, B is magnetic induction and L is the interaction length.For glass (crown) at a temperature of 18◦C, the Verdet constant is V = 2.68 × 10−5 deg/Gauss· mm.

13.3 Applications of BPM to photonic devices

The development of new types of components for WDM applications requires accuratemodelling before any fabrication attempts. Over the years many approaches and numericalmethods have been developed. They can generally be divided into two main groups [8]: time-harmonic (monochromatic CW operation) and general time-dependent (pulse operation).The important role in numerical analysis of those components is played by BPM.

There are many specific applications of the BPM to modelling different aspects of pho-tonic devices or circuits, such as passive waveguiding devices [9], channel-dropping filters[10], multimode waveguide devices [11], polarization splitters [12], multimode interferencedevices [13] and many more.


−5

0

5

010

2030

400

0.5

1

1.5

2

xz

Fig. 13.9 Propagation of Gaussian pulse in a waveguide by BPM.

−5

0

5

0

20

400

0.5

1

1.5

xz

Fig. 13.10 Propagation of Gaussian pulse in a tapered waveguide by BPM.

For a recent summary on integrated optics consult the recent book by Hunsperger [14].In this section we show the results of applications of the BPM developed earlier to the

simplest two waveguiding structures which are the building blocks of WDM devices. Here,we consider a rib waveguide and a tapered waveguide. In Figs. 13.9 and 13.10 we showpropagation of a Gaussian pulse and also profiles of the refractive indices. The MATLABcode used to generate those figures is provided in the Appendix, Listings 13A.1 and 13A.2.

To avoid reflections at the boundaries, transparent boundary conditions were imple-mented.

325 Appendix 13A

13.4 Projects

1. Function y junction.m provided in the Appendix can be used to create a Y-junction.Use this function and other BPM routines to develop a program which will propagatea Gaussian pulse in the Y-junction. Analyse various configurations from the literature.Consult Refs. [15] and [16].

2. Implement the method of finding propagation constants using BPM. Consult Refs. [17]and [18].

3. Write MATLAB code to analyse propagation of Gaussian pulses in a coupled rib wave-guide. Determine coupling length. Consult Refs. [19] and [20].

4. Analyse a fibre Bragg grating filter using BPM.



Listing 13A.1 Program bpm wg.m. Describes propagation of a Gaussian pulse in awaveguide using BPM.

% File name: bpm_wg.m

% Driver function which propagates Gaussian pulse

% in a strip waveguide using BPM with transparent boundary conditions

clear all

x_0 = 1; % center of Gaussian pulse





Table 13.1 List of MATLAB functions for Chapter 13.Listing Function name Description

13A.1 bpm wg.m Gaussian pulse in a waveguide with TBC13A.1.1 wg struct.m Constructs waveguiding structure used by bpmwg.m13A.1.2 step.m Propagates BPM solution along one step13A.2 bpm taper.m Gaussian pulse in a tapered waveguide with TBC13A.2.1 taper struct.m Constructs tapered structure used by bpmt aper.m13A.3 plot y.m Plots Y-junction13A.3.1 y junction.m Function defines Y-junction


N_x=128; % points on x axis


h=5*Delta_x; % propagation step


plot_index=zeros(N_x,N_z); % storage for plotting ref. index

plot_field=zeros(N_x,N_z); % storage for plotting field


x = x’;

E=exp(-((x - x_0)/w_0).^2); % initial Gaussian field

ref_index = wg_struct(x); % structure is uniform along z-axis

n_eff = 1.40; % assumed value of prop. constant

%

z = 0; z_plot = zeros(N_z); for r=1:N_z

z = z + h;

z_plot(r) = z + h;

plot_index(:,r)=wg_struct(x);

plot_field(:,r)=abs(E).^2;

E = step(Delta_x,k_0,h,ref_index,n_eff,E);

end;

%


plot(x,plot_index(:,k),’LineWidth’,1.2)

plot(x,plot_field(:,k),’LineWidth’,1.5)

hold on

end pause close all

%



plot3(x,y,plot_index(:,k),’LineWidth’,1.2)

plot3(x,y,plot_field(:,k),’LineWidth’,1.5)

hold on

end grid on xlabel(’x’)

ylabel(’z’) % along propagation direction

pause close all

Listing 13A.1.1 Program wg struct.m. Constructs a waveguiding structure.

function ref_index = wg_struct(x)

% Construction of the waveguide structure used by BPM

width = 2.0; % film width

n_c = 1.48; % refractive index of cover

n_s = 1.49; % refractive index of substrate

n_f = 1.52; % refractive index of film

ref_index = n_s*(x<0)+n_f*((x>=0)&(x<=width))+n_c*(x>width);

327 Appendix 13A

Listing 13A.1.2 Function step.m is slightly modified from the function of the same nameused in Chapter 12.

% File name: step.m

function E_new = step(Delta_x,k_0,h,ref_index,n_eff,E_old)

% Function propagates BPM solution along one step

N_x = size(E_old,1); % determine size of the system

last_term = k_0^2*Delta_x^2*(ref_index.^2 - n_eff^2);

prefactor = 1/(2*k_0*n_eff*Delta_x^2);

main = ones(N_x,1) - 0.5*last_term; above = ones(N_x-1,1); below =

above;


%

L_plus = eye(N_x) + 0.5i*h*P; % step forward

L_minus = eye(N_x)-0.5i*h*P; % step backward

%

%---- Implementation of boundary conditions

%

pref = 0.5i*h/(2*k_0*Delta_x^2);

k=1i/Delta_x*log(E_old(2)/E_old(1)); if real(k)<0

k=1i*imag(k);

end;

left = pref*exp(1i*k*Delta_x); % left correction for next step

L_plus(1) = L_plus(1)+left; L_minus(1) = L_minus(1)-left;

%

k=-1i/Delta_x*log(E_old(N_x)/E_old(N_x-1)); if real(k)<0

k=1i*imag(k);

end;

right = pref*exp(1i*k*Delta_x); % right correction for nest step

L_plus(N_x) = L_plus(N_x) + right; L_minus(N_x) = L_minus(N_x) -

right;

%

E_new = L_minus\L_plus*E_old; % determine new solution

Listing 13A.3 Function plots a Y-junction.

% File name: plot_y.m

% Plots y-junction

clear all

L_x=10.0; % computational window along x-axis

N_x=100; % number of steps along x-axis


L_z = 20; N_z = 100;

h = L_z/N_z; % step size


z=0;

%

for i=1:N_z

z = z + h;

z_plot(i) = z + h;

x = linspace(-L_x,L_x,N_x);

plotting(:,i)=y_junction(z,x,N_x,N_z);

end

%

for k = 1:10:N_z % choosing 3D plots every 10-th step


plot3(x,y,plotting(:,k),’.-’,’LineWidth’,1.0)

hold on

end xlabel(’x’); grid on pause close all

Listing 13A.3.1 Function defines a Y-junction.

function index = y_junction(z,x,N_x,N_z)

% Construction of the Y-junction structure used by BPM

% z - coordinate along propagation direction

% x - perpendicular coordinate

% Structure is described in only 2D, so we do not need cover

%

w = 2.0; % film width

n_s = 1.0; % refractive index of substrate

n_f = 1.52; % refractive index of film

a = 10; b = 20;

% Construction of Y-junction

slope = 0.1;

%

%index = zeros(N_x,N_z);

upper_1 = w/2 + slope*z; upper_2 = slope*z; lower_1 = - slope*z;

lower_2 = -w/2- slope*z;

%

n_1 = n_s*(x<-w/2)+n_f*((x>=-w/2)&(x<=w/2))+n_s*(x>w/2);

n_2 = n_s*(x<=lower_2)+n_f*((x>=lower_2)&(x<=lower_1))+...

n_s*((x>=lower_1)&(x<=upper_2))+n_f*((x>=upper_2)&(x<=upper_1))+...

n_s*(x>upper_1);

index = ((0<z)&(z<=a))*n_1+((a<z)&(z<=b))*n_2; end

329 References

References

[1] G. Keiser. Optical Fiber Communications. Third Edition. McGraw-Hill, Boston, 2000.[2] E. S. Koteles. Integrated planar waveguide demultiplexers for high density WDM

applications. In R. T. Chen and L. S. Lome, eds., Wavelength Division Multiplexing,pages 3–32. SPIE, 1999.

[3] B. Chomycz. Planning Fiber Optic Networks. McGraw-Hill, New York, 2009.[4] G. P. Agrawal. Fiber-Optic Communication Systems. Second Edition. Wiley, New

York, 1997.[5] K. O. Hill, Y. Fujii, D. C. Johnson, and B. S. Kawasaki. Photosensitivity in optical fiber

waveguides: applications to reflection filter fabrication. Appl. Phys. Lett., 32:647–9,1978.

[6] R. Kashyap. Fiber Bragg Gratings. Academic Press, San Diego, 1999.[7] S.-L. Chuang. Physics of Optoelectronic Devices. Wiley, New York, 1995.[8] R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert. Numerical techniques for

modeling guided-wave photonic devices. IEEE J. Select. Topics Quantum Electron.,6:150–62, 2000.

[9] L. Eldada, M. N. Ruberto, R. Scarmozzino, M. Levy, and R. M. Osgood. Laser-fabricated low-loss single-mode waveguiding devices in GaAs. J. Lightwave Technol.,10:1610, 1992.

[10] M. Levy, L. Eldada, R. Scarmozzino, R. M. Osgood, P. S. D. Lin, and F. Tong.Fabrication of narrow-band channel-dropping filters. IEEE Photon. Technol. Lett.,4:1378, 1992.

[11] I. Ilic, R. Scarmozzino, R. M. Osgood, J. T. Yardley, K. W. Beeson, and M. J. Mc-Farland. Modeling multimode-input star couplers in polymers. J. Lightwave Technol.,12:996–1003, 1994.

[12] M. Hu, J. Z. Huang, R. Scarmozzino, M. Levy, and R. M. Osgood. Tunable Mach-Zehnder polarization splitter using heigh-tapered Y-branches. IEEE Photon. Technol.Lett., 9:773–5, 1997.

[13] D. S. Levy, Y. M. Li, R. Scarmozzino, and R. M. Osgood. A multimode interference-based variable power splitter in GaAs-AlGaAs. IEEE Photon. Technol. Lett., 9:1373–5, 1977.

[14] R. G. Hunsperger. Integrated Optics. Theory and Technology. Sixth Edition. Springer,New York, 2009.

[15] M. H. Hu, J. Z. Huang, R. Scarmozzino, M. Levy, and R. M. Osgood. A low-loss andcompact waveguide Y-branch using refractive-index tapering. IEEE Photon. Technol.Lett., 9:203–5, 1997.

[16] T. A. Ramadan, R. Scarmozzino, and R. M. Osgood. Adiabatic couplers: design rulesand optimization. J. Lightwave Technol., 16:277–83, 1998.

[17] G. Lifante. Integrated Photonics. Fundamentals. Wiley, Chichester, 2003.[18] H. J. W. M. Hoekstra. On beam propagation methods for modelling in integrated

optics. Optical and Quantum Electronics, 29:157–71, 1997.


[19] C. Xu. Finite-Difference Techniques for Simulation of Vectorial Wave Propagation inPhotonic Guided-Wave Devices. PhD thesis, University of Waterloo, 1994.

[20] H.-P. Nolting, J. Haes, and S. Helfert. Benchmark tests and modelling tasks. InG. Guekos, ed., Photonic Devices for Telecommunications, pages 67–109. Springer,Berlin, 1999.

14 Optical link

In the present chapter we will combine some of the methods developed earlier to createthe simple point-to-point optical simulator which represents the simplest photonic system.It involves the transmitter, optical fibre and receiver. Some issues of how to quantify thequality of transmission in such a system will also be reviewed. Performance evaluation andtradeoff analysis are the central issues in the design of any communication system. Usingonly analytical methods, it is practically impossible to evaluate realistic communicationsystems. One is therefore left with computer-aided techniques.

In the last 10–15 years the design of photonic systems has moved from the back-of-the-envelope calculations to the use of sophisticated commercial simulators, see for example,products advertised by Optiwave, like OptiSystem [1], by RSoft Design Group the OpticalCommunication Design Suite [2] and by VPI Photonics line of products [3], to just namea few important players. They contain sophisticated physical models and allow for rapidassessment of new component technologies in the system under design.

Around 1995 the design of optical communication systems (operating over mediumdistances) would involve only a balance of power losses and pulse spreading. Later on,the demand on billion-dollar systems required complex analysis during the design process.This in turn created the need for sophisticated simulators.

Computer simulations can quickly provide answers to several important questions essen-tial to every engineer designing optical communication system, like: what repeater spacingis needed for a given bit rate, or what is the required power generated by a transmitter?

There exists extensive literature on the subject of modelling and simulation of fibre opticcommunication systems. Early extensive work on the description and calculation of thetransmission properties of optical fibres for communications was published by Geckeler[4]. There were several books published by van Etten and van der Plaats [5], Einarsson [6],Liu [7], Lachs [8], Keiser [9] and Palais [10], to list a few.

Literature published in journals is also extensive. Some of the relevant journal publica-tions are: [11], [12], [13], [14], [15], [16], [17], [18], [19], [20]; that includes statisticaldesign of long optical fibre systems is [21]. A general review intended for wide audiencewas written by Lowery [22].

14.1 Optical communication system

A generic optical link consisting of a transmitter, optical fibre and receiver was shown inthe Introduction, Fig. 1.1. In that figure we showed a transmitter which employs a laser

331

332 Optical link

Laser

Transmitter Channel Receiver

Photodetector Filter Sampler Thresholddetector

BERestimation

{Ak}{Bk}

Hf (ω)

Fig. 14.1 More realistic optical link.

AmpO-E E-O

Fiber FibreAmplifier

Optical receiver

Photonic regime Photonic regimeElectronic regime

Optical transmitter

Fig. 14.2 Principle of regeneration of optical signals. O-E means optical to electronic; E-O means electronic to optical.

diode to generate light, a channel which is formed by an optical fibre and a receiver whosemain element is a photodetector. Actual systems are more complicated.

In Fig. 14.1 we show a block diagram of a digital lightwave system (after Lima et al.[23]). Several detailed elements of the receiver are displayed. Some of them were alreadydiscussed in the earlier chapters. In the figure we also show notation for signals used tocharacterize operation of the particular elements.

Sequence Ak denotes the data values which are independent and identically distributed,Hf (ω) is the transfer function of the optical fibre and Bk is the output data sequence createdby the receiver.

When light is transmitted over long distances, optical pulses degenerate due to variouseffects discussed previously. As a result, maximum transmission distance is limited. Inorder to increase that distance, repeaters are installed. Typical structure of the repeater isschematically shown in Fig. 14.2. An optical pulse is converted into electrical form, thenamplified and regenerated and then converted back to light.

Such a process is costly and has also other limitations. Intensive research is conductedto have this process done entirely in the optical domain without conversion to an electronicregime.

Regenerators are devices which regenerate signals by removing noise and distortion afteramplification. The process is normally possible only in digital systems. It serves 2R or 3Rpurposes. In general, the following terminology has been established:

• 1R (Re-amplification only) – signal is amplified only.• 2R (Re-amplification and Re-shaping). Re-shaping corrects a pulse’s shape.

333 Design of optical link

Table 14.1 Typical values of losses for various components.

Wavelength [nm] Losses [dB km−1]

850 1.811300 0.351310 0.341380 0.401550 0.19

Table 14.2 Typical values of losses for various components.

Type of element Losses [dB]

Connector 0.1–0.2Splice losses 0.3

• 3R (Re-amplification, Re-shaping and Re-timing). Re-timing corrects the time drift inan optical pulse.

As a result of the regeneration process, clean digital pulses are created. The distancebetween repeaters varies, typically 50–70 km.

14.2 Design of optical link

There is a vast amount of literature on the design of optical communications links. Some ofthe useful papers and books are: Carnes et al. [24], Pepeljugoski and Kuchta [25], Iannoneet al. [26], Binh [27], Mortazy et al. [28], Zheng et al. [29], Lowery et al. [30], Sabella etal. [31], de Melo et al. [32].

The design of the optical link involves several interrelated variables which describeoperating characteristics of transmitter, fibre and receiver. The key practical requirementsare: transmission distance, data rate and bit error rate.

The simplest approach to the design of optical link is based on the evaluation of powerbudget and rise time budget [33]. Power budget analysis is needed to ensure that the systemwill work over the proposed distance with given components. The rise times of the source,fibre and the detector will determine the bandwidth available for transmission.

In the tables we will summarize losses in typical optical components. Losses ofa single-mode fibre from Corning Glass Works at various wavelengths are summa-rized in Table 14.1 [33]. Typical values of losses of other components are provided inTable 14.2.

334 Optical link

Connectors Splice

MarginReceiversensitivity

Connectorlosses

Fibrelosses

Splicelosses

Transmitteroutput

Distance fromtransmitter

Linklosses

Transmitter Receiver

Connectors

Fig. 14.3 Illustration of power budget.

14.2.1 Power budget analysis

In the analysis of power budget one determines the total allowed optical power loss betweenthe transmitter and the receiver. Typical components which contribute to losses are: fibre,connector, splices, couplers and splitters. Typical values of losses in standard componentsare summarized in Table 14.2.

In Fig. 14.3 we show a typical power budget which illustrates loss of power over existingcomponents as a function of distance from transmitter. In our example losses are createdwithin connectors, splices and within an optical fibre.

In calculating power budget one considers passive and active components of the link.Passive loss is made up of fibre loss, connectors, splices, couplers and splitters, amongothers. An example of a system power budget is illustrated in Table 14.3.

In this example, one has an excess power margin of 29 dB − 28 dB = 1 dB (total 7 dBwith the included system margin). As a general rule, the link loss margin should be greaterthan 3 dB to allow for link degradation over time, ageing of transmitters, etc. If duringoperation cables are accidentally cut, excess margin is needed to accommodate splices forrestoration.

14.2.2 Rise time budget

The rise time budget is needed to determine whether the link will operate at the requiredbit rate. The bit rate is mostly limited by the dispersion. As explained earlier, dispersion

335 Design of optical link

Table 14.3 Illustration of power budget determination.

TransmitterOutput power −13 dBm

ReceiverRx sensitivity −42 dBm

Margin 29 dBm

System lossFibre (3.5 × 5 km) 17.5 dBConnector (1 dB × 2) 2 dBSplicing (0.5 dB × 5) 2.5 dBSystem margin 6 dB

Total 28 dB

is responsible for pulse broadening as it travels through the fibre. Those effects introducerise time due to the material dispersion (τmat ) and modal dispersion (τmod). Additionally,the response of the transmitter and the receiver must also be included. Therefore, total linkrise time of the system is determined by the rms sum of all the rise times [33]:

τsys =√√√√ N∑

i=1

τ 2i =

√τ 2

tx + τ 2mat + τ 2

mod + τ 2rx (14.1)

For two main data coding schemes, namely RZ and NRZ, a general rule is that the systemrise time should be determined as

τsys,RZ ≤ 0.35

B, RZ (14.2)

τsys,NRZ ≤ 0.70

B, NRZ (14.3)

where B is the bit rate. The rise times of the optical transmitter and receiver are knownfrom the manufacturer. The rise time for multimode step index fibre is determined as [33](subscript im stands for intermodal)

τim = n1�

cL (14.4)

For material dispersion

τmat � 85L�λps (λ0 ∼ 850 nm)

0.5L�λps (λ0 ∼ 1300 nm)

� 20L�λps (λ0 ∼ 1500 nm)

(14.5)

where L is in km and �λ in nm.

336 Optical link

Table 14.4 Illustration of rise-time budget.

Component Rise time (ns)

System budget (NRZ) 1.75

Light source 1.0Fibre 0.23Photodetector

Transit time 0.5Circuit 1.3Total 1.4

System rise time 1.75 1.75

For single-mode fibres

τ f � D · L · �λ

� 2 · L · �λ (λ0 ∼ 1300 nm, λz ∼ 1300 nm)

� 16 · L · �λ (λ0 ∼ 1550 nm, λz ∼ 1300 nm)

� 2 · L · �λ (λ0 ∼ 1550 nm, λz ∼ 1550 nm)

(14.6)

with D � 2 ps/km·nm at 1300 nm and ∼ 16 ps/km·nm at 1550 nm with fibres withλz ∼ 1300 nm. Also D � 2 ps/km·nm at 1550 nm for fibres with λz ∼ 1550 nm. Here �λ

is the source spectral width, L is fibre length, λ0 is the operating wavelength and λz is thezero dispersion wavelength.

Typical rise time budget calculations are summarized in Table 14.4 [10].

14.3 Measures of link performance

There are various requirements which determine the performance of optical networks. Oneof them is network security, which will not be discussed here.

The simplest optical network is formed by a point-to-point optical link. Performance ofsuch a link can be determined by the amount of information received at the acceptable levelof errors [34]. Loss of information can be caused by a number of factors, of which sometypical ones are (a) decrease in the amplitude of the received signal and (b) loss of timing.

Signal quality is determined by analysing the so-called eye diagram and also signal-to-noise ratio (SNR). The main parameter determining the quality of a digital link is the biterror rate (BER). The smaller the bit error rate, the better the link transmission quality.Typical error rates for optical fibre telecommunication systems range from 10−9 to 10−12

[35]. They depend on the SNR at the receiver. BER was discussed in Chapter 10 dealingwith receivers. Here, we only discuss the eye diagram.

337 Measures of link performance

Transmitter

Trigger line

Receiver

Fibre

Test system

Bit patterngenerator

Trigger Data out

Oscilloscope

Vertical Trigger

Fig. 14.4 Schematic of an experimental equipment setup for making eye diagrammeasurements.

0 0 0

0 0

0 0

0

0 0

0

0

1

1

1

1

111

11

11

1

Fig. 14.5 Construction of an eye diagram for 3-bits.

14.3.1 Eye diagram

The eye diagram technique is a simple and convenient method of diagnosing problemswith data systems. The eye diagram is generated in the time domain using an oscilloscopeconnected to the demodulated data, see Fig. 14.4. Pseudorandom data are generated andapplied to the test system. After being transmitted through the test system, the data areapplied to the vertical input of an oscilloscope. The oscilloscope is triggered at everysymbol period (here using data from a separate line) or fixed multiple of symbol periods.As a result, the signal is retraced and superimposed on the same plot. Construction of aneye diagram for a 3-bit long sequence of data is shown in Fig. 14.5.

The result is an overlap of consecutive received symbols which form an eye pattern on theoscilloscope. The eye diagram provides a lot of information about the system’s performanceor about performance of its elements.

A created sequence of Gaussian pulses with random component for a predefined logicalcombination is shown in Fig. 14.6. For this sequence, the eye diagram is shown in Fig. 14.7.MATLAB code used in the generation of those results is shown in Appendix, Listing 14A.1.

338 Optical link

0 0.5 1 1.5 2 2.5 3× 10−7

0

0.2

0.4

0.6

0.8

1

time [s]

Fig. 14.6 Sequence of six Gaussian pulses with a random term. The logical data sequence is: 010101.

0 1 2 3 4 5× 10−8

0

0.2

0.4

0.6

0.8

1

Fig. 14.7 Generated eye diagram for the sequence of Gaussian pulses shown in Fig. 14.6. Eye opening is located in the middle ofall pulses.

14.4 Optical fibre as a linear system

In this section we consider the optical fibre modelled as a linear system and illustratepropagation of signals in such a system.

When an optical pulse propagates through optical fibre, it exhibits attenuation, delayand distortion. Delay can be handled through appropriate synchronization. Attenuationof a particular fibre determines the ultimate distance between transmitter and receiver.Distortion causes pulse broadening known as dispersion; it can produce situations when

339 Optical fibre as a linear system

(a)

(b)

h(t)Fibre s (t)in

s (t)in

s (t)in

s (t)outs (t)out

s (t)out

0 L z

tt = 0

t

Fig. 14.8 Illustration of pulse transmission in an optical fibre represented as a linear system described by the impulse responsefunction h(t). (a) Schematic of optical fibre and its representation as a linear system, (b) pulse broadening duringpropagation.

one cannot distinguish between pulses representing different logical information due totheir overlap.

Attenuation and dispersion determine practical limits on the information transmissionperformance of optical communication systems. Additional distortions, including noise,are produced in the receiver.

Propagation of optical signal of frequency ω through fibre of length L having attenuationcoefficient α is schematically illustrated in Fig. 14.8. In a first approximation, the opticalfibre is considered as a linear system with an impulse response h(t). Its Fourier transformH (ω) known as a transfer function is

H (ω) =∫ +∞

−∞h(t) e−iωt dt (14.7)

The fibre is not an ideal system. Therefore during the passage, the properties of the trans-mitted pulse described by its time function s(t) or its spectral function S(ω) are distorted,see Fig. 14.8. (Spectral function S(ω) and the time function s(t) are related by the FTrelation.)

In the frequency domain, the pulse propagation is described by the transfer functionH (ω) as

H (ω) = Sout (ω)

Sin(ω)= e−α(ω)−iφ(ω) (14.8)

where α(ω) is the system’s attenuation and φ(ω) its phase. Sout (ω) and Sin(ω) are FT ofoutput and input time functions of a propagating pulse.

Some of the examples of spectral functions H (ω) which are used to model pulse spectraand filter functions will now be summarized. These functions are easy to program. All aredimensionless, normalized low-pass functions approaching unity at ν = 0 and zero for

340 Optical link

ν −→ ∞ (we use the relation ω = 2πν, where ω is angular frequency and ν is frequency).They are real functions and symmetric with respect to ν = 0. The functions are [36]:

a) Gaussian pulse

HG(ν) = exp(−πν2τ 2

)(14.9)

b) Rectangular pulse of duration τ

HRP(ν) = sin (πντ )

πντ(14.10)

c) Raised cosine spectrum

HRC(ν) = cos2(πντ

2

)= 1

2(1 + cos πντ ) (14.11)

d) Spectrum of a trapezoidal pulse of duration τ and rise time tr

HT P(ν) = sin (πντ )

πντ· 1

2(1 + cos πνtr) (14.12)

e) Spectrum with Nyquist slope

HN (ν) = 1

2

[1 − sgn (1 − 2ντ ) ((1 − |2ντ |)N − 1)

](14.13)

where N is the parameter which controls roll-off of this function.f) Receiver filter that equalizes

HE (ν) = 1 + cos πντ

2 exp(−πν2τ 2Q

) (14.14)

We will now summarize a description of the three main elements of an optical link:transmitter, fibre and receiver.

14.5 Model of optical link based on filter functions

In a typical practical situation the optical system transmits pulses. As they propagate fromthe transmitter to the receiver, they broaden and also their slopes get distorted. A presenceof noise in all of a system’s components also adds to the modification of propagating pulse.

We start with a simple analysis where the whole system is modelled by a single function.Here, we only consider a rectangular pulse.

14.5.1 Test analysis for a rectangular pulse

In this section we present the analysis of propagation of a rectangular pulse. Although nota very realistic situation, it serves as a test.

In Fig. 14.9 on the left we show the input rectangular pulse. It propagates through thesystem modelled as a rectangular filter (shown on right). The output after propagation isshown in Fig. 14.10. MATLAB code in shown in the Appendix, Listing 14A.2.

341 Model of optical link based on filter functions

−40 −20 0 20 400

1

2

3

4

5

6

time (a.u.)

Puls

e am

plitu

de

−100 −50 0 50 1000

0.5

1

1.5

Frequency (a.u.)

Am

plitu

de

Fig. 14.9 Rectangular pulse (left) and rectangular filter.

−40 −20 0 20 400

1

2

3

4

5

6

Time (a.u.)

ampl

itude

Fig. 14.10 Rectangular pulse after regeneration.

PDMG

S (ω)1S (ω)2

Transmitter Fibre Receiver

H (ω)2

2s(t)

S (ω)0

P0

Eτ

0H (ω)0

0

EE0

1 H (ω)

1

1

τ τ

Fig. 14.11 Main elements of the transmission system.

Next, we build a more realistic model of the optical link, which is done in the followingsection. We summarize simple models for all the main components that constitute opticalfibre system and then develop a program which allows for the analysis of pulse propagationin such a system.

Our approach is based on the work by Geckeler [36]. Main parts of transmission systemare shown in Fig. 14.11. Here, E0 is the energy of the optical pulse produced by the

342 Optical link

Table 14.5 Parameters of typical generated pulses.

Parameter Symbol Value

Bit rate R0 50 Mbits/s−1

Power P0 0.5 mWNormalized pulse width T0 0.7

transmitter, τ0 is the width of that pulse and H0(ω) is the spectral function of the producedoptical pulse. Detailed discussion of all the three elements now follows.

14.5.2 Transmitter

It is usually a semiconductor laser which can be directly modulated by changing biascurrent or by employing an external modulator. The main characteristics of light source are:wavelength, output power, modulation speed, spectral width, noise. For a directly modulatedsemiconductor laser, the performance is determined by frequency chirping (discussed inChapter 7).

Here we assume a particular optical pulse as produced by a transmitter. Optical pulseswere discussed in Chapter 6. Of particular importance are rectangular and Gaussian pulses.

Pulse produced by the transmitter in the time domain is described by the function p0(t).Its Fourier spectrum is

S0(ω) =∫ +∞

−∞p0(t) e− jωt dt

Fourier spectrum is related to spectral function H0(ω) as

S0(ω) = p0,max · τ0 · H0(ω)

Parameters of the generated pulse are provided in Table 14.5. They refer to a system witha bit rate R0 = 50 Mbits/s−1 and an LED transmitter operating at 1.3 µm wavelength witha maximum launched power of 0.5 mW.

14.5.3 Fibre

The frequency spectrum of propagating pulse after it leaves fibre is obtained by multiplyingpulse spectral function and fibre spectrum.

Pulse propagation in the fibre is described by the product of two functions: D1 whichdescribes losses and its transfer function H1(ω). Losses are represented by

D1 = E1

E0= 10− αL1

10dB (14.15)

The function H1(ω) is a normalized filter function approximately expressed as a Gaussianlow-pass filter:

H1(ω) = e− 14π

πω2τ 21 (14.16)

343 Model of optical link based on filter functions

Table 14.6 Parameters of typical fibre.


Losses α 3 dBLength L1 15 kmMode coupling length Lc 10 kmBandwidth-length BL 500 MHz· kmFibre parameter τ1 Eq. (14.17)

The parameter τ1 can be determined from the measured bandwidth B1 of a fibre. Fibreparameters are listed in Table 14.6.

The fibre parameter τ1 is related to the bandwidth as

τ1 = 1

2B1(14.17)

Bandwidth is modelled after Geckeler [36] as

B1 = BL

(1

L1+ 1

3Lc

)(14.18)

where Lc is the mode coupling length and BL is the bandwidth-length product. The ap-proximation (14.18) is valid for 0 < L1 < 3Lc. The parameters BL and Lc are empiricalparameters. Their values are summarized in Table 14.6.

14.5.4 Receiver

An optical receiver is described by the low-pass filter transfer function

H2(ω) = 1

2

[1 + cos

(1

2ωτ2

)]at |ω| ≤ 2π

τ2(14.19)

with the parameter τ2.The signal spectrum at the receiver output is determined as [4]

S2(ω) = iT · T · M · R · G · H0(ω) · H1(ω) · H2(ω) (14.20)

where M is the multiplication factor of an avalanche photodiode (if used), G is its amplifi-cation factor and

iT = P0,max · τ0

T· D1

ηe

hν(14.21)

is the mean photocurrent of a pulse in a time slot of duration T .The parameter of the receiver is provided in Table 14.7.

14.5.5 Implementation of link model

The above approach has been implemented following work by Geckeler [36] and [4]. MAT-LAB code is shown in Appendix, Listing 14A.3. Typical results are shown in Fig. 14.12.

344 Optical link

Table 14.7 Parameter of a typical receiver.


Receiver parameter τ2 0.7

0 2 4 6 8−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Frequency (a.u.)

Fig. 14.12 Spectrum of rectangular pulse before (dotted line) and after propagation (solid line).

The signal spectrum generated by the transmitter is sin(πντ0)/(πντ0), which representsrectangular pulse of duration τ0. Input pulse is shown as a dotted line, the output signalafter the receiver as a solid line. More details were provided by Geckeler in Refs. [36] and[4].

14.6 Problems

1. Analyse spectral functions summarized in Section 14.4. Plot the time dependence ofeach function. Evaluate Fourier transform and plot frequency spectra.

2. Analyse Gaussian pulse using the filter approach.3. Analyse transmission of a Gaussian pulse by a single-mode fibre system. Determine

the output pulse width as a function of the input pulse width.

14.7 Projects

1. Write MATLAB code to construct an eye diagram for the program link.m.2. Add noise component to program link.m. Analyse the effect of noise on a system’s

performance.

345 Appendix 14A



14A.1 bits gen eye.m Generates pattern of Gaussian pulses and eye diagram14A.2 rectangular.m Rectangular pulse before and after filtering14A.3 link.m Simulation of optical link (rectangular pulse)



Listing 14A.1 Function bits gen eye.m Function generates a 6-bits long pattern of Gaus-sian pulses with random terms and eye diagram.

% File name: bits_gen_eye.m

% Purpose:

% Generates 6-bits long pattern of Gaussian pulses with random terms

% Based on generated sequence, eye diagram is created

%

clear all

bits = [0 1 0 1 0 1]; % def. of bit’s sequence

T_period = 50d-9; % pulse period [s]

t_zero = 20d-9; % center of incident pulse [s]

width = 6d-9; % width of the incident pulse [s]

% Generation of current pattern corresponding to bit pattern

I_p = 0;

N_div = 50; % number of divisions within each bit interval

t = linspace(0, T_period, N_div); % the same time interval is

% generated for each bit

g = exp(-0.5*((t_zero - t)/width).^2); % def. of a single Gaussian pulse

%

if bits(1)==0, I_p_1 = rand(1)*(1- g); % generates first bit

elseif bits(1)==1, I_p_1 = rand(1)*g;

end

temp = I_p_1;

number_of_bits = length(bits);

for k = 2:number_of_bits % generation of remaining bits

if bits(k)==1

A = rand(1)*g;

elseif bits(k)==0, A = rand(1)*(1- g);

end

A = [temp,A];

346 Optical link

temp = A;

end

temp_t = t;


t = linspace(0, T_period, N_div);

t = [temp_t,(k-1)*T_period+t];

temp_t = t;

end

%

h = plot(t,A,’LineWidth’,1.5);

xlabel(’time [s]’,’FontSize’,14); % size of x label


pause

close all

%

%=============== Eye diagram ============================

% put all bit plots on the first one

t_eye = linspace(0, T_period, N_div);

hold on

for m = 1:number_of_bits

A_temp = A((1+(m-1)*N_div):(m*N_div));

e = plot(t_eye,A_temp);

set(e,’LineWidth’,1.5); % thickness of plotting lines


end

pause

close all

Listing 14A.2 Function rectangular.m. Function which shows a rectangular pulse beforeand after filtering.

% File name: rectangular.m

% Analysis of rectangular pulse after filtering

%

clear all

hwidth = 10; % half-width of a pulse

time_step = 0.001; % time step

range = 40;

t = -range:time_step:range; % time range

%------ Creation of rectangular pulse -------

y = (5/2)*(sign(t+hwidth)-sign(t-hwidth));

plot(t,y,’LineWidth’,1.5);

xlabel(’time (a.u’,’FontSize’,14);

ylabel(’Pulse amplitude’,’FontSize’,14);


axis([-range range 0 6]);

347 Appendix 14A

pause

% %----- Fourier transform of a rectangular pulse -------

y_shift = fftshift(fft(y));

N=length(y_shift);

n=-(N-1)/2:(N-1)/2;

%------ Definitions of different filters ------------------

filter = (1/2)*(sign(n+100)-sign(n-100));

%filter = exp(-pi*n*0.0001);

%filter = 0.1*cos(pi*n*0.005).^2;

%filter = (1 + cos(0.05*n))/2;

plot(n,filter,’LineWidth’,1.5);

axis([-3*range 3*range 0 1.5]);

xlabel(’Frequency (a.u.)’,’FontSize’,14);

ylabel(’Amplitude’,’FontSize’,14);


pause

%-------- Regeneration of signal using filtered output ----------

y_reg=filter.*y_shift; % Filter original frequency spectrum

y_reg_inv=ifft(y_reg); % Inverse Fourier transform

plot (t,abs(y_reg_inv),’LineWidth’,1.3);

axis([-range range 0 6]);

xlabel(’Time(a.u.)’,’FontSize’,14);

ylabel(’amplitude’,’FontSize’,14);


pause

close all

Listing 14A.3 Function link.m. Function models propagation of a rectangular pulse inan optical link.

% File name: link.m

% Optical link simulations

% Follow code on p.133 of S. Geckeler (1983) paper

% No model of a transmitter

% No noise

%

clear all

%----------- Pulse produced by transmitter ----------------------

R_0 = 50d5; % Bit rate of binary digital signal, 50 Mbit/s

P_0 = 5d-4; % Max power of optical pulse, 0.5 mW

T_0 = 0.7; % Normalized pulse duration = tau_0/T

%------------- Fibre parameters -------------------------

L_1 = 15; % Fibre length, 15 km

A_L = 3.0; % Attenuation coefficient, 3 dB/km

D_1 = 10^(-A_L*L_1/10); % Power ratio

P_1 = P_0*T_0*D_1/2; % Mean optical power at the end of fibre

348 Optical link

B_L = 5d8; % Bandwidth-length product, 500 MHz km

L_C = 10; % Coupling length of a multimode fibre, 10 km

B_1 = B_L/L_1 + B_L/(3*L_C); % Fibre bandwidth

T_1 = R_0/(2*B_1);

%-------------- Receiver parameters -------------------------

T_2 = 0.7;

%============== System evaluation =======================================

% Output signal spectrum

D_F = pi/(20*T_2);

N = 32;

f = zeros(1,N);

f_A = zeros(1,N);

f_B = zeros(1,N);

f_C = zeros(1,N);

H_F = zeros(1,N);

H_F(1) = 1.0;

for j = 1:N

f(j) = j*D_F;

f_A(j) = sin(f(j)*T_0)/(f(j)*T_0); % transmitter

f_B(j) = exp(-f(j)^2*T_1*T_1/pi); % fibre

f_C(j) = (1/2)*(1 + cos(f(j)*T_2)); % receiver

H_F(j) = f_A(j)*f_B(j)*f_C(j);

end

%

plot(f,f_A,’.’,f,H_F,’LineWidth’,1.5) % Plot signal after propagation

xlabel(’Frequency (a.u.)’,’FontSize’,14);


pause

s = ifft(H_F);

s = s/max(s);

ff = linspace(0,2/3,N);

%

plot(fftshift(abs(s)),’LineWidth’,1.5);

xlabel(’time’,’FontSize’,14);

ylabel(’time output function’,’FontSize’,14);


grid

pause

close all

References

[1] Optiwave, www.optiwave.com/, 3 July 2012.[2] RSoft, www.rsoftdesign.com/, 3 July 2012.

349 References

[3] VPI photonics, www.vpiphotonics.com/, 3 July 2012.[4] S. Geckeler. Optical Fiber Transmission System. Artech House, Inc., Norwood, MA,

1987.[5] W. van Etten and J. van der Plaats. Fundamentals of Optical Fiber Communications.

Prentice Hall, New York, 1991.[6] G. Einarsson. Principles of Lightwave Communications. Wiley, Chichester, 1996.[7] M. M.-K. Liu. Principles and Applications of Optical Communications. Irwin,

Chicago, 1996.[8] G. Lachs. Fiber Optic Communications. Systems, Analysis, and Enhancements.

McGraw-Hill, New York, 1998.[9] G. Keiser. Optical Fiber Communications. Third Edition. McGraw-Hill, Boston, 2000.

[10] J. C. Palais. Fiber Optic Communications. Fifth Edition. Prentice Hall, Upper SaddleRiver, NJ, 2005.

[11] D. G. Duff. Computer-aided design of digital lightwave systems. IEEE J. Select. AreasCommun., 2:171–85, 1984.

[12] P. J. Corvini and T. L. Koch. Computer simulation of high-bit-rate optical fibertransmission using single-frequency lasers. J. Lightwave Technol., 5:1591–5, 1987.

[13] G. P. Shen, and T.-M. Agrawal. Computer simulation and noise analysis of the systemperformance at 1.5 µm single-frequency semiconductor lasers. J. Lightwave Technol.,5:653–9, 1987.

[14] A. F. Elrefaie, J. K. Townsend, M. B. Romeiser, and K. S. Shanmugan. Computersimulation of digital lightwave links. IEEE J. Select. Areas Commun., 6:94–104,1988.

[15] J. C. Cartledge and G. S. Burley. The effect of laser chirping on lightwave systemperformance. J. Lightwave Technol., 7:568–73, 1989.

[16] K. Hinton and T. Stephens. Modeling high-speed optical transmission systems. IEEEJ. Select. Areas Commun., 11:380–92, 1993.

[17] K. B. Letaief. Performance analysis of digital lightwave systems using efficient com-puter simulation techniques. IEEE Trans. on Communications, 43:240–51, 1995.

[18] A. Lowery, O. Lenzmann, I. Koltchanov, R. Moosburger, R. Freund, A. Richter,S. Georgi, D. Breuer, and H. Hamster. Multiple signal representation simulation ofphotonic devices, systems, and networks. IEEE J. Select. Topics Quantum Electron.,6:282–96, 2000.

[19] S. H. S. Al-Bazzaz. Simulation of single mode fiber optics and optical communicationcomponents using VC++. IJCSNS International Journal of Computer Science andNetwork Security, 8:300–8, 2008.

[20] L. N. Binh. MATLAB simulink simulation platform for photonic transmission sys-tems. I.J. Communications, Network and System Sciences, 2:97–117, 2009.

[21] T. J. Batten, A. J. Gibbs, and G. Nicholson. Statistical design of long optical fibersystems. J. Lightwave Technol., 7:209–17, 1989.

[22] A. J. Lowery. Computer-aided photonics design. Spectrum, 4:26–31, 1997.[23] R. A. A. Lima, M. C. R. Carvalho, and L. F. M. Conrado. On the simulation of digital

optical links with EDFAs: an accurate method for estimating BER through Gaussianapproximation. IEEE J. Select. Topics Quantum Electron., 3:1037–44, 1997.

350 Optical link

[24] H. Carnes, R. Kearns, and E. Basch. Digital optical system design. In E. E. B.Basch, ed., Optical-Fiber Transmission, pages 461–86. Howard W. Sams and Co.,Indianapolis, 1987.

[25] P. K. Pepeljugoski and D. M. Kuchta. Design of optical communications data links.IBM J. Res. and Dev., 47:223–37, 2003.


[27] L. N. Binh. Optical Fiber Communications Systems. CRC Press, Boca Raton, 2010.[28] E. Mortazy and M. K. Moravvej-Farshi. A new model for optical communication

systems. Optical Fiber Technology, 11:69–80, 2005.[29] H. Zheng, S. Xie, Z. Zhou, and B. Zhou. Simulation of optical fiber transmission

systems using SPW and an HP workstation. SIMULATION, 71:312–15, 1998.[30] A. J. Lowery and P. C. R. Gurney. Two simulators for photonic computer-aided design.

Applied Optics, 37:6066–77, 1998.[31] R. Sabella and P. Lugli. High Speed Optical Communications. Kluwer Academic

Publishers, Dordrecht, 1999.[32] C. F. de Melo Jr, C. A. Lima, L. D. S. Alcantara, R. O. dos Santos, and J. C. W.

A. Costa. A Simulink toolbox for simulation and analysis of optical fiber links. InJ. Javier Sanchez-Mondragon, ed., Sixth International Conference on Education andTraining in Optics and Photonics, volume 3831, pages 240–51. SPIE, 2000.

[33] G. Ghatak and K. Thyagarajan. Introduction to Fiber Optics. Cambridge UniversityPress, Cambridge, 1998.

[34] L. D. Green. Fiber Optic Communications. CRC Press, Boca Raton, 1993.[35] G. Keiser. Optical Fiber Communications. Fourth Edition. McGraw-Hill, Boston,

2011.[36] S. Geckeler. Modelling of fiber-optic transmission systems on a desk-top computer.

Siemens Res. and Dev. Reports, 12:127–34, 1983.

15 Optical solitons

In this chapter we concentrate on optical solitons and their propagation in optical fibre.They are pulses of a special shape and owe their existence due to the presence of dispersionand nonlinearity in optical fibre. They are able to propagate ultra-long distances whilemaintaining their shape.

Here, we provide some basic knowledge for understanding the underlying physicalprinciples of soliton creation and propagation. We generalize the theory of linear pulsesdeveloped in Chapter 5 and include the nonlinear part of polarization. We will then derivea nonlinear equation which describes propagation of solitons in optical fibre.

Soliton propagation in an optical fibre was first demonstrated at Bell Laboratories byL. F. Mollenauer, R. S. Stolen and J. P. Gordon. We refer to the book by Mollenauer andGordon [1] for the description of basic principles and a brief history of solitons.

Some other relevant books on optical solitons and also on applications in optical com-munications are [2], [3], [4], [5], [6].

15.1 Nonlinear optical susceptibility

Solitons exist due to nonlinearity and dispersion. Dispersion was discussed earlier in thebook. Here, we will concentrate on nonlinear effects. Optical responses including nonlineareffects are described as [7]

P(t) = ε0{χ(1)E(t) + χ(2)E2(t) + χ(3)E3(t) + · · · } (15.1)

≡ P(1)(t) + P(2)(t) + P(3)(t) + · · ·

where we have expressed polarization P(t) as a power series in the field strength E(t). Thequantities χ(1), χ (2), χ (3) are known as susceptibilities; χ(1) is a linear susceptibility andχ(2), χ (3) are known as the second order- and third-order nonlinear susceptibilities.

For a typical solid-state system χ(1) is of the order of unity whereas χ(2) is of the order of1/Eat and χ(3) of the order of 1/E2

at , where Eat = e/(4πε0a20) is the characteristic atomic

electric field strength and a0 = 4πε0�2/me2 is the Bohr radius of the hydrogen atom.

Explicitly [7]

χ(2) � 1.94 × 10−12m/V

χ(3) � 3.78 × 10−24m2/V 2

351

352 Optical solitons

Formal expression for the third-order susceptibility is [7]

Pi(ω0 + ωn + ωm) = ε0D∑

jkl

χ(3)

i jkl (ω0 + ωn + ωm, ω0, ωn, ωm)

× Ej(ω0) Ek(ωn) El (ωm)

where i, j, k, l refer to the Cartesian components of the fields and the degeneracy factor Drepresents the number of distinct permutations of the frequencies ω0, ωn, ωm.

χ( j) ( j = 1, 2, . . .) is the j-th order susceptibility. The linear susceptibility χ(1) con-tributes to the linear refractive index n0 (real and imaginary parts; the imaginary partbeing responsible for attenuation). The second-order susceptibility χ(2) is responsible forthe second harmonic generation. For SiO2 the second-order nonlinear effect is negligiblesince SiO2 has the inversion symmetry. Therefore optical fibres normally do not show thesecond-order nonlinear effects.

The third-order susceptibility χ(3) is responsible for the lowest order nonlinear effectsin optical fibres. Generally, it manifests itself as the change in the refractive index withoptical power or as a scattering phenomenon. It is linked with the optical Kerr effect, four-wave mixing, third-harmonic generation, stimulated Raman scattering, etc. An excellentdiscussion of third-order optical susceptibilities has been reported by Hellwarth [8].

Assuming linear polarization of propagating light and neglecting tensorial character ofχ

(3)

i jkl , one finds the following relation for the nonlinear polarization:

PNL(ω) = 3ε0χ(3) (ω = ω + ω − ω) |E(ω)|2E(ω)

Total polarization, which consists of linear and nonlinear parts, is written as

P(ω) = ε0χ(1)E(ω) + 3ε0χ

(3)|E(ω)|2E(ω)

= ε0χeff E(ω)

The effective susceptibility is field dependent as

χeff = χ(1) + 3χ(3)|E(ω)|2

and it is linked to refractive index as

n = 1 + χ(3) ≡ n0 + n2I (15.2)

Here I denotes the time-averaged intensity of the optical field. We start with the discussionof the main features of nonlinear effects.

15.2 Main nonlinear effects

15.2.1 Kerr effect

It was discovered by J. Kerr in 1875. He found that a transparent liquid becomes dou-bly refracting (birefringent) when placed in a strong electric field. Generally, Kerr effect

353 Derivation of the nonlinear Schrodinger equation

E(z,t)

(a) (b)

t

A(z,t)

E(z,ω)~

Δω

ω0

ω

Fig. 15.1 Illustration of the propagating modulated pulse (a) and its spectrum (b).

describes situations where refractive index depends on electric field as

n(ω, |E|2) = n0 (ω) + n2 (ω) E|2

Here, n2 is known as Kerr coefficient and it is related to susceptibility as

n2 (ω) = 3

4πχ(3)

xxx

for a linearly polarized wave in the x direction. For silica its value is approximately 1.3 ×10−22 m2/V2. Kerr effect originates from the non-harmonic motion of electrons bound inmolecules. Consequently, it is fast effect, the response time of the order of 10−15s.

15.2.2 Stimulated Raman scattering

Scattering phenomena are responsible for Raman and Brillouin effects. During thosescatterings, the energy of the optical field is transferred to local phonons: in Ramanscattering optical phonons are generated whereas in the Brillouin scattering the acousticphonons.

15.3 Derivation of the nonlinear Schrodinger equation

Solitons in optical fibres are described by the so-called nonlinear Schrodinger (NSE)equation, which will be now derived. In the derivation we use the concept of the Fourierspectrum of the propagating pulse, see Fig. 15.1.

A medium where solitons propagate exhibits Kerr nonlinearity. In such a medium,refractive index depends on intensity of electric field I(t) given by Eq. (15.2), againreproduced here:

n(t) = n0 + n2 · I(t) (15.3)


where [7]

I(t) = 2n0ε0c|A(z, t)|2 (15.4)

Here A(z, t) is the slowly varying envelope connected to the optical pulse described by theoptical pulse E(z, t) as, see Fig. 15.1:

E(z, t) = A(z, t) ei(ω0t−β0z) (15.5)

Fourier transform of the optical field is [3], [9]

E(z, t) =∫ +∞

−∞dω E(z, ω) ei(ωt−βz) (15.6)

where E(z, ω) is the Fourier spectrum of the pulse, β propagation constant and ω0 thefrequency at which the pulse spectrum is centered (also known as carrier frequency), seeFig. 15.1.

For quasi-monochromatic pulses with �ω ≡ ω − ω0 � ω0, it is useful to expandpropagation constant β(ω) in a Taylor series:

β(ω) = β0 + β1 · (ω − ω0) + 1

2β2 · (ω − ω0)

2 + �βNL (15.7)

where we have neglected higher order derivatives. Here �βNL = n2k0I is the nonlinearcontribution to the propagation constant.

Substitute expansion (15.7) into Eq. (15.6):

E(z, t) = e−iβ0z∫ +∞

−∞d(�ω) E(z, ω) exp

(iωt − iβ1z�ω − 1

2iβ2z�ω2 − i · z · �βNL

)= ei(ω0t−β0z)

∫ +∞

−∞d(�ω) E(z, ω0 + �ω)

× exp

(it�ω − iβ1z�ω − 1

2iβ2z�ω2 − i · z · �βNL

)≡ ei(ω0t−β0z) A(z, t)

where we have introduced

A(z, t) =∫ +∞

−∞d(�ω) E(z, ω0 + �ω)

× exp

(it�ω − iβ1z�ω − 1

2iβ2z�ω2 − i · z · �βNL

)≡∫ +∞

−∞d(�ω) E(z, ω0 + �ω) eig(z,t) (15.8)

Our next step is to obtain a differential equation describing evolution of the amplitudeA(z, t) from Eq. (15.8) which is in the integral form. To do this, one needs to take partial

355 Derivation of the nonlinear Schrodinger equation

derivatives of Eq. (15.8). One obtains

∂A(z, t)

∂t=∫ +∞

−∞d(�ω) E(z, ω0 + �ω) i�ω eig(z,t)

∂2A(z, t)

∂t2=∫ +∞

−∞d(�ω) E(z, ω0 + �ω) (i�ω)2 eig(z,t)

∂A(z, t)

∂z=∫ +∞

−∞d(�ω) E(z, ω0 + �ω)

(−iβ1�ω − 1

2iβ2�ω2 − i · �βNL

)eig(z,t)

When evaluating time derivatives, we have assumed in the above that I(t) does not dependon time. Addition of the above combination of derivatives produces

∂A(z, t)

∂z+ β1

∂A(z, t)

∂t− i

1

2β2

∂2A(z, t)

∂t2

=∫ +∞

−∞d(�ω) E(z, ω0 + �ω)

×[(

−iβ1�ω − 1

2iβ2�ω2 − i · �βNL

)+ β1i�ω − i

1

2β2(i�ω)2

]× eig(z,t)

(15.9)

The term in the bracket is, [· · · ] = −i · �βNL = −in2k0I . Eq. (15.9) thus gives

∂A(z, t)

∂z+ β1

∂A(z, t)

∂t− i

1

2β2

∂2A(z, t)

∂t2

=∫ +∞

−∞d(�ω) E(z, ω0 + �ω) (−in2k0I ) eig(z,t)

= −in2k0I

∫ +∞

−∞d(�ω)E(z, ω)eig(z,t)

≡ −in2k0I A(z, t)

when using (15.8). The final equation describing solitons is therefore [3], [10]

∂A(z, t)

∂z+ β1

∂A(z, t)

∂t+ i

1

2β2

∂2A(z, t)

∂t2= iγ |A(z, t)|2 A(z, t) − α

2A(z, t) (15.10)

where we have defined nonlinear coefficient γ (after [3]) as

γ = 2πn2

λAeff(15.11)

Here Aeff is the effective core area.Our interests here lie in the pulse evolution during propagation and not in the time of pulse

arrival. We can therefore simplify the above equation by transforming it to a coordinatesystem which moves with group vg. In this moving frame, new time T and new coordinateZ are

Z = z (15.12)

T = t − β1z


To obtain the transformed equation, we must evaluate derivatives with respect to newvariables as follows:

∂A

∂t= ∂A

∂T

∂T

∂t+ ∂A

∂|Z∂Z

∂t= ∂A

∂T

since ∂T∂t = 1 and ∂Z

∂t = 0. From the above one finds

∂2A

∂t2= ∂2A

∂T 2

Using the above results, one has

∂A

∂z= ∂A

∂T

∂T

∂z+ ∂A

∂|Z∂Z

∂z= −β1

∂A

∂T+ ∂A

∂|ZThe last result is used in Eq. (15.10) to replace ∂A

∂t . The transformed equation is

∂A

∂z+ i

1

2β2

∂2A

∂T 2− iγ |A|2A + 1

2αA = 0 (15.13)

where in the final step we replaced Z by z. It is known as a nonlinear Schrodinger equation(NSE).

To further analyse NSE, we will introduce two characteristic lengths describing dispersion(LD) and nonlinearity (LNL). Those are defined as

LD = T 20

|β2| = T 20 2πc

|D|λ2(15.14)

and

LNL = 1

γ P0(15.15)

where P0 is the peak power of the slowly varying envelope A(z, T ) and T0 is a temporalcharacteristic value of the initial pulse, which is often defined as full width half maximum(the pulse 3 dB width). Those two lengths characterize how far a pulse must propagate toshow the respective effect. Physically, LD is the propagation length at which a Gaussianpulse broadens by a factor of

√(2) due to group velocity dispersion (GVD).

GVD dominates pulse propagation in fibres whose length L is L � LNL and L ≥ LD.In such situation. the nonlinearity in NLSE can be ignored and the equation can be solvedanalytically. Nonlinear effects dominate in fibre where L � LD and L ≥ LNL. In this limitthe dispersion term can be ignored.

Typical values of parameters used in the simulations are summarized in Table 15.1.For numerical analysis normalize variables

U = 1√P0

A and τ = T

T0(15.16)

The width parameter T0 is related to the full-width at half-maximum (FWHM) intensity ofthe input pulse. Specifically

Ts = 2T0ln(1 +√

(2)) ≈ 1.763T0 (15.17)

357 Split-step Fourier method

Table 15.1 Typical parameters used in simulation of solitons.

Parameter Symbol Value Unit

wavelength λ 1.55 µmnonlinear coeff. γ 1.3 1/ kmWGVD β2 15 × 10−24 s2/kmwidth parameter T0 100 pspeak power P0 0.15 mWlosses α 0.01 dB/km−1

chirp parameter c 1.2 dimensionlesssoliton period z0 1047.2 km

After simple algebra, Eq. (15.13) takes the form

∂U

∂z− i

sign(β2)

2LD

∂2U

∂τ 2+ i

1

LNL|U |2U + 1

2αU = 0 (15.18)

Another normalized form of the Schrodinger equation exists in the literature. We obtain itin the lossless case, i.e. with α = 0. To derive it, normalize the z coordinate as follows:

ξ = z

LD(15.19)

After a few algebraic steps, one obtains

∂U

∂ξ− i

sign(β2)

2

∂2U

∂τ 2+ iN2|U |2U = 0 (15.20)

where N is known as soliton order and is defined as

N2 = LD

LNL= γ P0T 2

0

|β2| (15.21)

The last popular form of the NLSE is found by introducing u as

u = NU =(

γ P0T 20

|β2|)1/2

A (15.22)

Eq. (15.20) then takes the form

∂u

∂ξ− i

sign(β2)

2

∂2u

∂τ 2− i|u|2u = 0 (15.23)

15.4 Split-step Fourier method

In this section we will discuss the numerical solution of the nonlinear Schrodinger equation(NSE) which describes propagation of optical solitons using the so-called split-step Fouriermethod (SSFM).


L

h

A1

Ai Ai+1

A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12

(a)

(b)

nonlinear

linearlinear

h/2 h/2

Ai+ 12

Fig. 15.2 Illustration of split-step Fourier method. (a) Division of optical fibre intoN regions (hereN = 11) of equal lengths.(b) Illustration of operation of linear and nonlinear operations at arbitrary segments.

The SSFM is a numerical technique used to solve nonlinear partial differential equationslike the NSE. The method relies on computing the solution in small steps and on taking intoaccount the linear and nonlinear steps separately. The linear step (dispersion) can be madein either frequency or time domain, while the nonlinear step is made in the time domain.The method is widely used for studying nonlinear pulse propagation in optical fibres. Formore detail see [11]

A nonlinear Schrodinger equation, Eq. (15.13) contains dispersive and nonlinear terms.To introduce SSFTM, write the NLSE equation in the following form:

∂A(z, T )

∂z= (

L + N)

A(z, T ) (15.24)

where

LA = −α

2− i

2β2

∂2A

∂T 2(15.25)

contains losses and dispersion in the linear medium and nonlinear term

NA = iγ |A|2A (15.26)

accounts for the nonlinear effects in the medium.The basis of the SSFM is to split a propagation from z to z + h (h is a small step) into

two operations (assuming that they act independently): during first step nonlinear effectsare included and in the second step one accounts for linear effects, see Fig. 15.2.

359 Split-step Fourier method

Formal solution of Eq. (15.24) over a small step h is thus

A(z + h, t) = eh(L+N)A(z, t) (15.27)

In the first-order approximation, the above formula can be written as

A(z + h, t) = ehL ehN A(z, t) + O(h2) (15.28)

The basis of this approximation is established by Baker-Hausdorf lemma [12], which is

eA eB = eA+B e12 [A,B] (15.29)

given that operators A and B commute with [A, B].The basis of the method is suggested by Eq. (15.28). It tells us that A(z + h, t) can be

determined by applying the two operators independently. The propagation from z to z + his split into two operations: first the nonlinear step and then the linear step assuming thatthey act independently. If h is sufficiently small, Eq. (15.28) gives good results.

The value of step h can be determined by assuming that the maximum phase shiftφmax = γ |Ap|2h, where Ap is the peak value of A(z, t) due to the nonlinear operator issmaller than the predefined value. Iannone et al. [2] reported that φmax ≤ 0.05 rad.

For a practical implementation of the SSFM, we need to establish practical expressionsfor dispersive and nonlinear terms. In the following we will therefore analyse the effect ofboth terms independently neglecting losses.

Let us analyse the effect of the dispersive term alone. For that we temporally switch offthe nonlinear term. After Fourier transform, the ‘linear equation’ becomes

∂A(z, ω)

∂z= − i

2ω2β2A(z, ω)

which has the solution

A(z, ω) = A(0, ω)e−iω2β2z/2

The action of the nonlinear term alone is described by the equation

∂A(z, t)

∂z= iγ |A(z, t)|2A(z, t)

The ‘natural’ solution is in the time domain. It produces

A(z, t) = A(0, t)eiγ |A|2A

15.4.1 Split-step Fourier transformmethod

The propagation medium (say, cylindrical optical fibre) is divided into small segments, eachof length h, see Fig. 15.2. Further, each individual segment of length h is subdivided intotwo of equal lengths. The linear operator operates over each subsegment in the frequencydomain, whereas the nonlinear operator operates only locally at the central point.


Operation of the linear operator L, Eq. (15.25) over first subsegment is done as follows:

ehL/2A(z, t) = F−1{

ehL/2 F {A(z, t)}}

(15.30)

i.e. one must Fourier transform original amplitude from time domain into frequency domain,apply linear operator L and then apply inverse Fourier transform to get the amplitude backto time domain.

The operation of nonlinear operator defined by Eq. (15.26) is as follows:

Ai+1/2,L(z, t) = Ai+1/2,R(z, t) eh N (15.31)

where Ai+1/2,L is the value of field amplitude at an infinitesimal point left from i+ 12 . Finally,

the operation of linear operator over second subsegment of length h2 is done exactly the

same way as over the first segment.To summarize, the method over each segment of length h consists of three steps:

step 1

⎧⎪⎨⎪⎩Ai(z, ω) = F {Ai(z, t)}Ai−(z, ω) = Ai(z, ω) · exp

(−i 12ω2β2h

)Ai−(z, t) = F−1

{Ai−(z, ω)

}step 2 Ai+(z, t) = Ai−(z, t) · exp

(iγ |A|2Ah

)step 3

⎧⎪⎨⎪⎩Ai+1(z, ω) = F {Ai+1(z, t)}Ai+1(z, ω) = Ai+1(z, ω) · exp

(−i 12ω2β2h

)Ai+1(z, t) = F−1

{Ai+(z, ω)

}where F indicates Fourier transform (FT) and F−1 inverse FT.

15.4.2 Symmetrized split-step Fourier transformmethod (SSSFM)

The simulation time of Eq. (15.28) essentially depends on the size of step h. To reducesimulation time, a new method was invented which allows us to use larger steps, h.

Mathematically, one uses the following second-order approximation:

A(z + h, t) = e12 hL exp

{∫ z+h

zN (z′)dz′

}e

12 hLA(z, t) + O(h3) (15.32)

In this approach one assumes that the nonlinearities are distributed over h, which is morerealistic. For small h, one can approximately evaluate∫ z+h

zN (z′)dz′ ≈ h

2

[N (z) + N (z + h)

]In the SSSFM algorithm Eq. (15.31) is replaced by

Ai+1/2,L(z, t) = Ai+1/2,R(z, t) eh[N (z)+N (z+h)] (15.33)

The above scheme requires iteration to find Ai+1 since it is not known at z + 12 h. Initially

N (z + h) will be assumed to be the same as N (z).

361 Numerical results

−10000

1000

0

500

10000

0.005

0.01

0.015

time (ps)distance (km)

Fig. 15.3 Evolution in time over one soliton period forN = 1 soliton.

−10000

1000

0

500

10000

0.005

0.01

0.015


Fig. 15.4 Evolution in time over one soliton period forN = 1 soliton with damping.

15.5 Numerical results

Based on the developed software, in this section we will report on numerical analysis ofsoliton propagation in optical fibre. The SSFM has been implemented and MATLAB codeis provided in Appendix, Listing 15A.1. We start with the single solitons.

15.5.1 Single solitons

In our analysis we used fibre and pulse parameters summarized in Table 15.1. The inputsoliton was considered to be in the form

Ain (ξ = 0, T ) = √A0 sech

(T

T0

)(15.34)

where N is the soliton order given by Eq. (15.21). In Fig. 15.3 we illustrated the propagationof N = 1 soliton without damping and in Fig. 15.4 the evolution of the same soliton withdamping (α = 0.01 dB km−1).


−10000

1000

0

500

10000

0.05

0.1

0.15

0.2


Fig. 15.5 Evolution in time over one soliton period forN = 3 soliton. Note soliton splitting near z0 = 0.5 and its recoverybeyond that point.

−10000

1000

0

500

10000

0.05

0.1

0.15

0.2


Fig. 15.6 Evolution in time over one soliton period forN = 3 soliton with damping.

A higher-order soliton with N = 3 is defined as

Ain (ξ = 0, T ) = N3√A0 sech

(T

T0

)(15.35)

Propagation of higher-order soliton with N = 3 is shown in Fig. 15.5 (no damping) and inFig. 15.6 (with damping). One can observe periodic evolution of the undamped soliton.

15.5.2 Chirped solitons

Here we analyse how chirp affects single soliton propagation. We assume the followinginput:

Ain (ξ = 0, T ) = √A0 sech

(T

T0

)exp

[−iC/2

(T

T0

)2]

(15.36)

363 Numerical results

−10000

1000

0

500

10000

0.005

0.01

0.015

0.02


Fig. 15.7 Evolution in time of a chirped soliton forN = 1without damping.

−10000

1000

0

500

10000

0.005

0.01

0.015


Fig. 15.8 Evolution in time of two interacting solitons without damping.

where C is the chirped parameter. Evolution of such soliton in the case of N = 1 andC = 1.6 is shown in Fig. 15.7. The pulse is initially compressed and then broadens.

15.5.3 Two interacting solitons

The effect of nonlinearity produces mutual interaction between soliton pulses if they arelaunched close together [13]. This interaction is important from a practical point of viewand also from a fundamental perspective related to soliton propagation. For example, it wasshown [13] that nonlinear interaction between solitons can result in a bandwidth reductionby a factor of 10. See also the review in [14].

First, we consider an interaction between two solitons of the same strength and N = 1.We assume the initial pulses of the form [15]

Ain (ξ = 0, T ) = √A0sech

(T

T0− 1

)+ √

A0sech

(T

T0+ 1

)(15.37)

The result of two interacting solitons is shown in Fig. 15.8.


TB

0 0011 1 1

solitons

Fig. 15.9 Stream of solitons.

15.6 A few comments about soliton-based communications

As mentioned at the beginning of this chapter, an important application of solitons is inthe transmission of information in optical fibre systems. Soliton pulses are stable whenthey propagate over long distances. Losses in fibres are an important limiting factor, so itbecomes necessary to compensate periodically for fibre loss. This can be done by usingEDFA.

In the transmission of information, solitons are considered like linear pulses. In a bitstream each soliton has its own bit slot and it represents logical one or zero, see Fig. 15.9.Here TB is the duration of the bit slot and it is related to bit rate B as

TB = 1

B= 2aT0 (15.38)

where T0 is the soliton width and 2a specifies the distance between neighbouring solitons.To prevent their interactions, the neighbouring solitons should be well separated.

Solitons are very well suited for long-haul communication because of their highinformation carrying capacity and the possibility of periodic amplification. However,soliton systems are still waiting for the full field deployment. The reader might con-sult recent publications on soliton-based optical communication systems [16], [17],[18], [19].

15.7 Problems

1. Implement a symmetrized SSFM based on Eq. (15.33).2. Generalize the implementation by including the third-order dispersion effect where

β3 �= 0. Analyse the influence of β3 on propagation of N = 1 and N = 3solitons.

3. Implement SSFM for a normalized form of NLSE given by Eq. (15.23). Using yourimplementation, conduct the analysis for N = 1 and N = 3 solitons.

4. Analyse the propagation of Gaussian pulses in fibres with nonlinearity using SSFM.Consider regular Gaussian and also chirped Gaussian pulses.

365 Appendix 15A



15A.1 ss f tm f ull.m Analysis of various types of solitons



Listing 15B.1 Program ssftw full.m. Here we conduct analysis of various solitons usingthe split-step Fourier transform method.

% File name: ssftm_full.m

% Split-step Fourier transfer method for the analysis of solitons

% Contains the following initial pulses

% N=1 soliton

% N=3 solitons

% Chirp N=1 soliton

% Two interacting solitons

% For the analysis of the appropriate case it should be uncommented

%

clear all

tic;

%===== Material parameters ===================

%alpha=0.01; % Fibre loss (dB/km)

alpha = 0.0;

alpha=alpha/(4.343); % Fibre loss (1/km)

gamma = 1.3; % Fibre nonlinearity [1/W/km]

beta_2=15d-24; % 2nd order disp. (s^2/km)

%====== Time parameters ===========

T =- 512e-12:1e-12: 511e-12;

delta_t=1e-12;

%======= Input pulse and its parameters ========

T_0 = 100d-12; % Initial pulse width (s)

P_0 = 0.15d-3; % Input power (Watts)

A_0 = sqrt(P_0);

L_D =((T_0^2)/(abs(beta_2))); % Dispersion length in [km]

L_NL = (1/(gamma*P_0));

h = 2d0; % step size [km]

z_plot = (0:h:((pi/2)*L_D)); % z-values to plot [in km]

n = 1; % controls stepping

z_0 = (pi/2)*L_D; % Soliton period [km]

N = 3; % Soliton number


%A = N^2*A_0*sech(T/T_0); % N=3 soliton

%A = A_0*sech(T/T_0); % N=1 soliton

C = 1.6; % chirp parameter

%A = A_0*sech(T/T_0).*exp((-1i*C/2).*(T/T_0).^2); % N=1 soliton with chirp

A = A_0*sech(T/T_0-1.6)+A_0*sech(T/T_0+1.6); % interacting solitons

%

L = max(size(A));

Delta_omega=1/L/delta_t*2*pi;

omega = (-L/2:1:L/2-1)*Delta_omega;

A_f = fftshift(fft(A)); % Pulse in frequency domain

for jj=h:h:z_0

A_f = A_f.*exp(-alpha*(h/2)-1i*beta_2/2*omega.^2*(h/2));

A_t =ifft(A_f); % Pulse in time domain

A_t = A_t.*exp(1i*gamma*((abs(A_t)).^2)*(h));

A_f = fft(A_t);

A_f = A_f.*exp(-alpha*(h/2)-1i*beta_2/2*omega.^2*(h/2));

A_t = ifft(A_f);

plotting(:,n) = abs(A_t);

n = n+1;

end

toc;

cputime=toc;

%

z_plot = h:h:z_0; % creates distance for plotting

length(z_plot)

for k = 1:80:length(z_plot) % Choosing 3D plots every 10-th step

y = z_plot(k)*ones(size(T));% Spread out along y-axis

plot3(T*1d12,y,plotting(:,k),’LineWidth’,1.5)

hold on

end

xlabel(’time (ps)’,’FontSize’,14);

ylabel(’distance (km)’,’FontSize’,14);


grid on

pause

close all

References

[1] L. F. Mollenauer and J. P. Gordon. Solitons in Optical Fibers. Fundamentals andApplications. Elsevier, Amsterdam, 2006.


[3] G. P. Agrawal. Fiber-Optic Communication Systems. Second Edition. Wiley, NewYork, 1997.

367 References

[4] Y. S. Kivshar and G. P. Agrawal. Optical Solitons. From Fibers to Photonic Crystals.Academic Press, Amsterdam, 2003.

[5] A. Hasegawa and M. Matsumoto. Optical Solitons in Fibers. Springer, Berlin, 2003.[6] A. Hasegawa and Y. Kodama. Solitons in Optical Communications. Clarendon Press,

Oxford, 1995.[7] R. W. Boyd. Nonlinear Optics. Third Edition. Academic Press, Amsterdam, 2008.[8] R. W. Hellwarth. Third-order optical susceptibilites of liquids and solids. Prog. Quant.

Electr., 5:1–68, 1977.[9] A. Yariv and P. Yeh. Photonics. Optical Electronics in Modern Communications. Sixth

Edition. Oxford University Press, New York, Oxford, 2007.[10] G. P. Agrawal. Nonlinear Fiber Optics. Academic Press, Boston, 1989.[11] G. P. Agrawal. Nonlinear Fiber Optics, Fourth Edition. Elsevier Science & Technology

Books, Boston, 2006.[12] R. L. Liboff. Introductory Quantum Mechanics. Addison-Wesley, Reading, MA, 1992.[13] P. L. Chu and C. Desem. Mutual interaction between solitons of unequal amplitudes

in optical fibre. Elect. Lett., 21:1133–4, 1985.[14] C. Desem and P. L. Chu. Soliton-soliton interactions. In J. R. Taylor, ed., Optical

Solitons – Theory and Experiment, pages 107–51, Cambridge University Press,Cambridge, 1992.

[15] Z. B. Wang, H. Y. Yang, and Z. Q. Li. The numerical analysis of soliton propagationwith split-step Fourier transform method. Journal of Physics: Conference Series,48:878–82, 2006.

[16] A. Hasegawa. Soliton-based optical communications: An overview. IEEE J. Select.Topics Quantum Electron., 6:1161–72, 2000.

[17] M. F. Ferreira, M. V. Facao, S. V. Latas, and M. H. Sousa. Optical solitons in fibersfor communication systems. Fiber and Integrated Optics, 24:287–313, 2005.

[18] R. Gangwar, S. P. Singh, and N. Singh. Soliton based optical communication. Progressin Electromagnetics Research, PIER, 74:157–66, 2007.

[19] N. Vijayakumar and N. S. Nair. Solitons in the context of optical fibre communications:a review. Journal of Optoelectronics and Advanced Materials, 9:3702–14, 2007.

16 Solar cells

In this chapter we provide the fundamentals of solar cell operations. We outline basicphysical principles, summarize a model based on equivalent circuit and discuss ways toincrease efficiency of single-junction solar cell by employing multijunction structures orby creating intermediate bands.

General introductory books describing solar cells are [1], [2] and [3]. Anderson [1] andRabl [2] provide a practical background in solar-energy conversion including techniquesfor estimating the solar radiation incident upon a collector, fundamentals of optics for solarcollectors, discussion of types of concentrators and economical analysis, among others.Moeller [3] concentrates on fundamental, physical and material aspects of semiconductorsfor photovoltaic energy conversion.

16.1 Introduction

World consumption of electric energy circa 2009 was around 12–13 TW [4]. This energywas created by several methods. Let us look briefly at two of them: nuclear power and solarenergy.

Assuming that a single nuclear plant produces 1 GW of power, creation of 10 TW ofpower would require 10 000 nuclear power plants. This huge number will certainly createsocial problems. Also, the uranium needed for those plants could be diminished in less than20 years.

As far as the solar energy is concerned, the Sun supplies more solar energy on Earthin one hour than we use globally in one year. To make another comparison, a footballfield covered with silicon solar cells with an efficiency around 17% at 1-Sun illuminationproduces approximately 500 kW of power. The USA uses about 3 TW, so to produce thatamount one will need about 6 million football fields. However, by increasing efficiencyof solar cells, for example by using multijunction devices and increasing concentration ofsolar energy, the area of semiconductor solar cells can be significantly reduced [5].

Those rough estimates indicate that solar energy can be a practical alternative as a sourceof electric energy, and also that increasing the efficiency of solar cells is important. Oneshould, however, remember that the incidence of solar energy is not uniform.

Solar irradiation spectrum (spectrum of solar energy) is shown in Fig. 16.1. This graphwill be important later when we discuss in detail multijunction cells. In the figure weshow spectral intensity, that is intensity of solar energy per unit wavelength. Above theEarth’s atmosphere it is denoted as AM0 (air-mass zero). The integrated (over the whole

368

369 Introduction

2.5

2.0

1.5

1.0

0.5

0.2 0.4 0.6 0.8 1.0

Wavelength [μm]

Ene

rgy

dist

ribu

tion

[kW

/m2

μm]

Black body radiation 6000 K

AM0 radiation

AM1.5 radiation

1.2 1.4 1.6 1.8 2.0

Fig. 16.1 Solar irradiation spectrum. Reprinted and adapted with permission from C. H. Henry, J. Appl. Phys. 51, 4494 (1980).Copyright 1980, American Institute of Physics.

hΘh0Atmosphere

Fig. 16.2 Definition of air-mass.

spectrum) intensity above the Earth’s atmosphere is constant and its approximate value is1.353 kW m2. It gives the total power flow through a unit area perpendicular to the directionof the Sun.

The Earth’s atmosphere significantly absorbs radiation from the Sun. In Fig. 16.1 theradiation at the Earth’s surface is labelled as AM1.5. In the figure several absorption bandsand lines due to various effects are shown. For comparison, the spectrum of black bodyradiation at the temperature of 6000 K is also shown. One can notice that the solar spectrumabove the Earth’s atmosphere (AM0) resembles black body radiation.

The air mass coefficient defines the direct optical path length through the Earth’s at-mosphere, expressed as a ratio relative to the path length vertically upwards, i.e. at thezenith.

Let us finish this section with an explanation of AM (air-mass) convention. Generally airmass m (AMm) is defined (see Kasap [6]) as the ratio of the actual radiation path h to theshortest path h0, m = h/h0 = cos �, see Fig. 16.2.

Changing the light intensity incident on a solar cell changes all solar cell parameters,including the short-circuit current, the open-circuit voltage, the efficiency and the impactof series and shunt resistances. The light intensity on a solar cell is measured in the numberof ‘suns’, where 1 sun corresponds to standard illumination at AM1.5, or 1 kW m2. For

370 Solar cells

sunlight

p-type

n-type

Fig. 16.3 A p-n junction operating as a solar cell.

example, a system with 10 kW m2 incident on the solar cell would be operating at 10 suns,or at 10X.

16.2 Principles of photovoltaics

The simplest solar cell (or photovoltaic (PV) cell) is a single p-i-n junction operatingunder forward bias. It is formed by using, e.g. properly doped silicon, see Fig. 16.3. It canproduce potential difference, thus acting as a battery. It contains a very narrow and heavilydoped n-region. It is covered with thin antireflecting coating to reduce sun reflection whichpenetrates p-n junction from n-side, see Fig. 16.4 for more details.

Short wavelengths penetrate only the n-region, longer wavelengths penetrate depletionregion of width w, and finally the longest wavelengths reach the p-region, see Fig. 16.5.

Photons of frequency ν having energies hν ≥ Eg, where Eg is the bandgap energy,generate electron-hole (e-h) pairs. Due to a built-in electric field E0 in the depletion region,electrons move towards metal contact in the n-region whereas holes travel in oppositedirection towards metallic contact in the p-region. Since the n-region is very narrow, mostof the photons are absorbed in the depletion region. Each generated electron increasescharge in the n-region by −e; similarly each hole makes the p-region more positive by +e.Thus potential difference is created between metallic electrodes.

A semiconductor can only efficiently convert photons into current with energies equal tothe bandgap. Photons with energies smaller than the bandgap are not absorbed, and photonswith higher energies reduce their energies to the bandgap energy by thermalization of thephotogenerated carriers; a process which involves losses.

The basic relation which provides the link between photons’ energy and their wavelengthis

λ[µm] = 1.24

E[eV ](16.1)

If both terminals of solar cell are shorted, the excess electrons on the n-side will startflowing through an external wire contributing to electrical current known as photocurrent,

371 Principles of photovoltaics

bottomcontact

top contact

incident light

p+ layer

p-layer

n-layer

antireflectioncoating

Fig. 16.4 Solar cell perspective.

+

–

Ev

EFc

Ec

Eg

AR coatingn-region p-region

Iphw

E

λ

0

Fig. 16.5 The principle of photocurrent generation in a solar cell in short circuit.

Iph, see Fig. 16.6. The short circuit conventional current Isc flows opposite to photocurrent.If Ilight is the light intensity, one can write

Isc = −Iph = −K · Ilight

where K is some (device dependent) constant.

372 Solar cells

Id – IphI =

R

V

Fig. 16.6 Solar cell driving an external load resistance R.

Iph

V

I

Voc

darkunder illumination

power

Fig. 16.7 Finding an operating point of a solar cell.

When the illuminated solar cell is loaded with resistance R, the conventional current I isI = Id − Iph, see Fig. 16.6, where Id is a forward diode current:

Id = I0

[exp

(eV

ηkBT

)− 1

](16.2)

and where I0 is a constant, V is the voltage across diode and η is known as diode fidelityfactor. It is equal to 1 for diffusion controlled and 2 for space charge layer recombinationcontrolled characteristics.

The I-V characteristic of solar cell is shown in Fig. 16.7. It is obtained from a darkdiode by shifting its I-V curve downward by Iph. The analytical expression for conventionalcurrent is

I = −Iph + I0

[exp

(eV

ηkBT

)− 1

](16.3)

With the load resistance R connected to the solar cell, see Fig. 16.6, the current-voltagerelation is

I = −V

R(16.4)

Operating point of solar cell with load resistance R is obtained by solving Eqs. (16.3) and(16.4). The graphical solution is illustrated in Fig. 16.8., where we have plotted Eqs. (16.3)and (16.4). Crossing of both lines determines operating point (V1, I1) of a solar cell.

The point at which a characteristic intersects the vertical current axis is known as a shortcircuit condition and the corresponding current as short-circuit current Isc. Similarly, thepoint at which I-V characteristic intersects the horizontal voltage axis is known as the open

373 Equivalent circuit of solar cells

Isc –Iph

V

IVoc

P (operating point)

=

Fig. 16.8 Finding an operating point of a solar cell.Voc is an open-circuit voltage.

Iph Id

I

Rs

RshV

Fig. 16.9 Equivalent circuit of a solar cell.

circuit condition. It defines the open-circuit voltage Voc, which is the maximum voltagewhich can be drawn from the solar cell. It corresponds to zero current.

Power delivered to the load R is Pload = I1 · V1, which is the area of the rectangle, seeFig. 16.7. The goal of the solar cell design is to maximize that power.

16.3 Equivalent circuit of solar cells

Over the years several methods of modelling solar cells have been introduced. The simplestapproach is based on equivalent circuit models. Some recently published works with anemphasis on using MATLAB are [7], [8], [9].

16.3.1 Basic model

The equivalent circuit of solar cell which includes parasitic effects is shown in Fig. 16.9.It consists of current source which generates photoelectric current Iph, ideal diode, shuntresistor and series resistor. Based on this model, the goal is to obtain current-voltage relation.From Kirchoff rule one obtains (from now on we reverse direction of flow of conventionalcurrent, which is a common practice)

I = Iph − Id (16.5)

where I is the cell total current and Id is the diode current. Combining the above andrelation (16.3), one finds

I = Iph − I0

[exp

(eVd

ηkBT

)− 1

](16.6)

374 Solar cells

This equation describes an ideal situation without parasitic effects. In a real solar cellthere exists leakage current flowing through shunt resistance Rsh. Potential drop across thedevice is represented by series resistance Rs. The inclusion of shunt resistance modifiessolar current I as

I = Iph − Id − Vd

Rsh(16.7)

The impact of series resistance is included as

Vd = V + I · Rs (16.8)

With the parasitic effects included, the solar current I takes the form

I = Iph − I0

[exp

(eVd

ηkBT

)− 1

]− V + I · Rs

Rsh(16.9)

The photocurrent Iph depends on the solar insolation and cell’s temperature and it is givenby [10], [11]

Iph = [Isc + KI

(Tc − Tre f

)]Sin (16.10)

where Isc is the cell’s short-circuit current at a 25◦ C and 1 kW m2, KI is the cell’s short-circuit current temperature coefficient, Tre f is the cell’s reference temperature and Sin is thesolar insolation in kW m2.

The expression for I0 is [10], [11]

I0 = IRS

(Tc

Tre f

)3/n

exp

⎡⎣ Egap

AkB

(1

Tre f− 1

Tc

)⎤⎦ (16.11)

where IRS is cell’s reverse saturation at a reference temperature and a solar radiation, Egap

is the bandgap energy of the semiconductor used in the cell and A is an ideal factor whichdepends on PV technology. Some typical values are: A = 1.2 for Si-mono, A = 1.3 forSi-poly and A = 1.5 for CdTe (after [10]).

16.3.2 Other models

There exist other models of solar cells which are based on a basic model. Here, we willonly mention two such models: the double exponential model and ideal PV cell model.

Double exponential model [12] is shown in Fig. 16.10. This model contains a lightgenerated current source, two diodes D1 and D2 and a series and parallel resistances. Themodel is derived from the physics of the p-n junction and can describe cells constructedfrom polycrystalline silicon.

Simple models. The shunt resistance Rsh is inversely related to shunt leakage current tothe ground. PV efficiency shows little sensitivity to the variation of Rsh and one can assumethat Rsh = ∞. The resulting model is similar to the basic one, shown in Fig. 16.9, withoutRsh. The current-voltage expression for this model is

I = Iph − I0

[exp

(e (V + I · Rs)

ηkBT

)− 1

](16.12)

375 Equivalent circuit of solar cells

Table 16.1 Parameters for equivalent circuit model of Si solar cell.

Description of parameter Symbol Value

Illumination Iph 10 mAeta η 1.5Current I0 3 × 10−6 mASeries resistance Rs 0 �, 20 �, 50 �

Shunt resistance Rsh 415 �

Iph

I

RsD2D1

Rsh V

Fig. 16.10 Double exponential model of a solar cell.

0 0.2 0.4 0.6 0.80

0.005

0.01

0.015

0.02

Cur

rent

(m

A)

Voltage (V)

RS = 50 Ω RS = 20 Ω RS = 20 Ω

Fig. 16.11 Current-voltage characteristics of an ideal solar cell based on Si for several values of series resistance.

An ideal PV cell model assumes no series loss and no leakage to ground; i.e. Rs = 0 andRsh = ∞. The equivalent circuit consists of only current source and one diode. Current isexpressed as

I = Iph − I0

[exp

(eV

ηkBT

)− 1

](16.13)

We conducted analysis using typical parameters, which are summarized in Table 16.1.Current-voltage characteristics for several values of series resistance are shown in Fig. 16.11and in Fig. 16.12 we presented current-voltage characteristics at three different values

376 Solar cells

0 0.2 0.4 0.6 0.80

0.005

0.01

0.015

0.02

Cur

rent

(mA

)

Voltage (V)

300K

350K

Fig. 16.12 Current-voltage characteristics of an ideal solar cell based on Si for several values of temperature.

of temperatures. MATLAB code used to generate those characteristics is shown in theAppendix, Listings 16A.1, 16A.2 and 16A.3. These are typical results as reported in theliterature.

16.4 Multijunctions

As determined by Shockley and Queisser [13], the maximum theoretical limit for a singlejunction solar cell is about 31%. In order to increase theoretical efficiency one needs toincrease number of p-n junctions in the cell. Henry [14] found that maximum theoreticalefficiency at a concentration of 1 sun is 31%. At a concentration of 1000 suns with the cellat 300 K, the maximum efficiencies are 37%, 50%, 56% and 72% for cells with 1, 2, 3 and36 p-n junctions (with different, properly chosen energy gaps), respectively.

Increase in efficiencies can be obtained by splitting solar spectrum into several partsand using different materials for conversion in various parts. This principle is illustrated inFig. 16.13 for a triple solar cell built from Ga0.49In0.51P(1.9 eV), Ga0.99In0.01As(1.4 eV) andGe(0.7 eV) (adopted from Dimroth [15]). Germanium is used mainly due to its robustnessand its low cost of production. Those different semiconductor materials are grown onone substrate. Tunnel junctions are used to form an electrical series connection of thesubcells.

Suitable materials must be chosen for each cell so that photons of appropriate wavelengthsare absorbed. The bandgap of each material is determined using Eq. (16.1). Energy bandgapEg must decrease from top cell to bottom cell in order not to absorb all photons by the topcell. Also, all layers of semiconductors must be lattice matched, so that there is no built-instrain.

The anti-reflective (AR) coating typically consists of several layers. Design of AR layerscan be performed using methods discussed in Chapter 3.

377 Multijunctions

0.2

0.5

1.0

1.5

2.0

2.5

0.40.6

0.81.0

1.21.4

1.61.8

wavelength (μm

)visible range

AM

0 radiation

Middlecell

Bottom

cellTop cell

Energy distribution (kW/ m2 μm)

Tunnel junction

AR coating

wavelength

long middle short

Top cell: GaInP

Middle cell: GaAs

Bottom cell: Ge

Ge substrate

Tunnel junction

Contact

Contact

Fig. 16.13 Triple-junction solar cell. (From F. Dimroth, High-efficiency solar cells from III-V compound semiconductors, Phys. Stat.Sol. (c) 3, 373–9 (2006). Copyright Wiley-VCH Verlag GmbH and Co. KGaA. Reproduced with permission.)

The presence of tunnel junctions is to provide low electrical resistance between subcells.Modelling of those layers specifically for the purpose of MJ solar cells has been describedrecently by Baudrit and Algora [16].

16.4.1 Quantum dots in multijunctions

The triple-junction subcells shown are connected in series and their voltages add up, butthe current flowing through the structure is determined by the smallest current produced bysubcells. Total power generated by the structure is greater than that of the single junctioncell.

The thickness of each subcell is determined by its absorption and the light power availablein the spectral range where the subcell operates. Referring to Fig. 16.13, the bottom cell(formed by Ge, including substrate) is much thicker than two upper subcells. Due to thisand also because it covers much larger spectral range, the Ge subcell produces much largercurrent compared to two upper subcells.

One of the possible solutions of reducing current of the Ge subcell and at the sametime increasing current of the GaAs subcell, is to introduce InAs-based quantum dots intothe GaAs subcell [17]. These quantum dots are designed to have an effective bandgapslightly smaller than that of GaAs. Therefore they can ‘steal’ current from bottom Gasubcell roughly by about 15%. Increasing also the thickness of top GaInP subcell to matchthe increased current of GaAs results in an overall current increase by about 15%. With

378 Solar cells

Ev

EIB

Ec

Fig. 16.14 Illustration of a concept of intermediate-band solar cell.

the minimal change in voltage over GaAs subcell, this concept resulted in an increase oftriple-junction conversion efficiency by about 13%.

16.4.2 Intermediate band solar cells (IBSC)

The concept of intermediate band solar cells (IBSC) was first described by Luque and Marti[18] and by Wolf [19]. The main idea was to create an additional band between conductionand valence bands, see Fig. 16.14. The additional intermediate band can be created usingseveral approaches, like, for example, minibands.

The standard cell is only able to absorb photons with energy equal to or greater thanEgap = Ec − Ev . Within this new scheme it is possible to have additional absorption viathe IB.

The electronic states in the IB should be accessible via direct transitions. Thereforephoton of energy EIV can transfer an electron from the valence band into IB whereasanother photon with energy ECI will be able to pump electrons from IB to conduction band.Thus the performance of solar cell is increased.

Solar cells based on this design are attractive because of their predicted photon conver-sion efficiency of up to about 60% [20]. The authors of Ref. [20] show that the orderedthree-dimensional arrays of quantum dots, i.e., quantum dot supracrystals, can be used toimplement the intermediate-band solar cell with the efficiency significantly exceeding theShockley-Queisser limit for a single junction cell. The increase is due to the utilization ofphotogenerated hot carriers which can produce higher voltages and higher photocurrents.

Modelling of such structures has been conducted using the CFD Research Corporation3D device simulator, NanoTCAD [21]. They performed accurate simulations of quantumdot solar cell performance and degradation due to effects of space radiation.

A recent summary of some aspects of intermediate-band solar cells was published inNature Photonics [22].

16.4.3 Role of simulations

Numerical simulations and modelling of solar cells play an important role. One particularexample has been mentioned in the preceding section. Optimization of modern structures,especially for multijunction or quantum dots by experimental trial-and-error methods, isdefinitely too costly. This opens the room for simulations.

379 Appendix 16A

The optical and electrical interactions between the numerous layers in a multi-junctionsolar cell are very complex. Over the years several groups have developed simulation modelsof various levels of sophistication. The simplest approach based on equivalent circuitsmodel was summarized in an earlier section. There were also many successful attempts todevelop microscopic models of various complexity. Full discussion of microscopic models,including those based on drift-diffusion approach, is beyond the scope of this book.

Also, a few commercial products are found on the market: Sentaurus from Synopsys,Atlas from Silvaco or Solar Cell utility from RSoft. Some recent examples of using com-mercial software to simulate and design solar cells are described in [23], [24] and [25].The authors of Ref. [25] describe details of using Sentaurus to simulate quantum well solarcells.

More sophisticated approaches to designing of quantum well solar cells using quantumtransport are also under development [26], [27]. Such approaches combine elements fromsemiconductor optics and quantum transport in nanostructures. A very fundamentalapproach based on non-equilibrium Green’s functions (NEGF) is discussed extensivelyin [26]. The developed approach treats absorption, transport and relaxation on equal foot-ing and within a framework based on non-equilibrium quantum statistical mechanics.

We finish this discussion by mentioning the role played by plasmonics in improvingproperties of photovoltaic devices, see a recent summary [28]. Plasmonics describe guidingand localizing light at the nanoscale smaller than the wavelength of light in free space.Using plasmonics allows for new designs of solar cells in which light is fully absorbed in asingle quantum well. Proposed solutions can result in improved absorption in photovoltaicdevices.



Listing 16A.1 Program solar Si.m. MATLAB program which plots I-V characteristics ofan ideal solar cell for several values of series resistance based on equivalent circuit model.

% File name: solar_Si.m

% Simulates ideal solar cell based on Si



16A.1 solar Si.m Generates I-V characteristics for several values of series resistance16A.2 current V.m Function used by solar Si.m and solar Si T.m16A.3 solar Si T.m Generates I-V characteristics for several values of temperature

380 Solar cells

clear all

eta = 1.5;

I_ph = 0.02; % illumination current, in mA

T = 330;

V_init = 0;

V_final = 1.0;

V_step = 0.01;

V=0;

hold on

for R_s = [0.01 20 50] % series resistance, in ohms

curr_total = fzero(@(curr) current_V(curr,V,T,eta,I_ph,R_s),0.2);

for V = V_init:V_step:V_final

curr = fzero(@(curr) current_V(curr,V,T,eta,I_ph,R_s),0.2);

curr_total = [curr_total, curr];

end

V_plot = V_init:V_step:V_final; % creation voltage values for plot

V_plot = [0, V_plot];

plot(V_plot, curr_total,’LineWidth’,1.5)


axis([0 V_final*0.8 0 I_ph*1.2]);

ylabel(’Current (mA)’,’Fontsize’,14)

xlabel(’Voltage (V)’,’Fontsize’,14)

text(0.65, 0.01, ’R_s = 0 \Omega’,’Fontsize’,14)



end

pause

close all

Listing 16A.2 Function current V.m. MATLAB function which creates an expression forcurrent in the equivalent circuit model.

function fun = current_V(I,V,T,eta,I_ph,R_s)

% Expression for current used in solar cell model based on equivalent

% circuit

q = 1.6d-19; % charge of electron

k_B = 1.38d-23; % Boltzmann constant

R_sh = 415; % shunt resistance in ohms

I_0 = 3d-9; % reverse saturation current of the diode

VV = V+I*R_s;

fun = I_ph - I - I_0*(exp(q*VV/(eta*k_B*T))-1) - VV/R_sh;

Listing 16A.3 Function solar Si T.m. MATLAB program which plots current-voltagecharacteristics of an ideal solar cell for three temperatures using the equivalent circuitmodel.

381 References

% File name: solar_Si_T.m

% Simulates effect of temperature for ideal solar cell based on Si

clear all

eta = 1.5;

I_ph = 0.02; % illumination current, in mA

R_s = 10; % series resistance, in ohms

V_init = 0;

V_final = 1.0;

V_step = 0.01;

V=0;

hold on

for T = [300 325 350] % temperature

curr_total = fzero(@(curr) current_V(curr,V,T,eta,I_ph,R_s),0.2);

for V = V_init:V_step:V_final

curr = fzero(@(curr) current_V(curr,V,T,eta,I_ph,R_s),0.2);

curr_total = [curr_total, curr];

end

V_plot = V_init:V_step:V_final; % creation voltage values for plot

V_plot = [0, V_plot];

plot(V_plot, curr_total,’LineWidth’,1.5)


axis([0 V_final*0.8 0 I_ph*1.2]);

ylabel(’Current (mA)’,’Fontsize’,14)

xlabel(’Voltage (V)’,’Fontsize’,14)

text(0.65, 0.005, ’300K’,’Fontsize’,14)

text(0.38, 0.01, ’350K’,’Fontsize’,14)

end

pause

close all

References

[1] E. E. Anderson. Fundamentals of Solar Energy Conversion. Addison-Wesley, Read-ing, MA, 1983.

[2] A. Rabl. Active Solar Collectors and Their Applications. Oxford University Press,New York, Oxford, 1985.

[3] H. J. Moeller. Semiconductors for Solar Cells. Artech House, Boston, 1993.[4] K. Tanabe. A review of ultrahigh efficiency III-V semiconductor compound solar

cells: multijunction tandem, lower dimensional, photonic uo/down conversion andplasmonic nanometallic structures. Energies, 2:504–30, 2009.

[5] G. S. Kinsey, R. A. Sherif, R. R. King, C. M. Fetzer, H. L. Cotal, and N. H. Karam.Concentrator multijunction solar cells for utility-scale energy production, 2005.unpublished.

382 Solar cells

[6] S. O. Kasap. Optoelectronics and Photonics: Principles and Practices. Prentice Hall,Upper Saddle River, NJ, 2001.

[7] F. M. Gonzalez-Longatt. Model of photovoltaic module in Matlab. 2do CongresoIberoamericano de Estudiantes de Ingenieria Electrica, Electronica y Computacion(II Cibelec, 2005), pages 1–5, 2005.

[8] R. K. Nema, S. Nema, and G. Agnihotri. Computer simulation based study of pho-tovoltaic cells/modules and their experimental verification. International Journal ofRecent Trends in Engineering, 1:151–6, 2009.

[9] M. Azab. Improved circuit model of photovoltaic array. International Journal ofElectrical Power and Energy Systems Engineering, 2:185–8, 2009.

[10] H.-L. Tsai, C.-S. Tu, and Y.-J. Su. Development of generalized photovoltaic modelusing Matlab/Simulink. Proceedings of the World Congress on Engineering andComputer Science, WCECS 2008, 2008.

[11] R. Hernanz, C. Martin, Z. Belver, L. Lesaka, Z. Guerrero, and E. P. Perez. Modellingof photovoltaic module. International Conference on Renewable Energies and PowerQuality, ICREPQ’10, 2010.

[12] J. A. Gow and C. D. Manning. Development of a photovoltaic array model for use inpower-electronics simulation studies. IEE Proc.-Electr. Power Appl., 146:193–200,1999.

[13] W. Shockley and H. J. Queisser. Detailed balance limit of efficiency of p-n junctionsolar cells. J. Appl. Phys., 32:510–19, 1961.

[14] C. H. Henry. Limiting efficiencies of ideal single and multiple energy gap terrestrialsolar cells. J. Appl. Phys., 51:4494–500, 1980.

[15] F. Dimroth. High-efficiency solar cells from III-V compound semiconductors. Phys.Stat. Sol.(C), 3:373–9, 2006.

[16] M. Baudrit and C. Algora. Tunnel diode modeling, including nonlocal trap-assistedtunneling: a focus on III-V multijunction solar cell simulation. IEEE Trans. ElectronDevices, 57:2564–71, 2010.

[17] B. Riel. Quantum dots: higher CPV efficiencies, same production cost. Future Pho-tovoltaics, May, 2010.

[18] A. Luque and A. Marti. Increasing the efficiency of ideal solar cells by photon inducedtransitions at intermediate levels. Phys. Rev. Lett., 78:5014–17, 1997.

[19] W. Wolf. Limitations and possibilities for improvement of photovoltaic solar energyconverters. Proc. IRE, 48:1246–63, 1960.

[20] Q. Shao, A. A. Balandin, A. I. Fedoseyev, and M. Turowski. Intermediate-band solarcells based on quantum dot supracrystals. Appl. Phys. Lett., 91:163503, 2007.

[21] A. I. Fedoseyev, M. Turowski, A. Raman, E. W. Taylor, S. Hubbard, S. Polly, S. Shao,and A. A. Balandin. Space radiation effects modeling and analysis of quantum dotbased photovoltaic cells. In E. W. Taylor and D. A. Cardimona, eds., Proc. of SPIE.Nanophotonics and Macrophotonics for Space Environments III, volume 7467, page746705. 2009.

[22] A. Luque and A. Marti. Towards the intermediate band. Nature Photonics, 5:137–8,2011.

383 References

[23] M. Hermle, G. Letay, S. P. Philipps, and A. W. Bett. Numerical simulation of tunneldiodes for multi-junction solar cells. Prog. Photovolt: Res. Appl., 16:409–18, 2008.

[24] S. P. Philipps, M. Hermle, G. Letay, F. Dimroth, B. M. George, and A. W. Bett.Calibrated numerical model of a GaInP-GaAs dual-junction solar cell. Phys. Stat.Sol. (RRL), 2:166–8, 2008.

[25] P. Kailuweit, R. Kellenbenz, S. P. Philipps, W. Guter, A. W. Bett, and F. Dimroth.Numerical simulation and modeling of GaAs quantum-well solar cells. J. Appl. Phys.,107:064317, 2010.

[26] U. Aeberhard. A microscopic theory of quantum well photovoltaics. PhD thesis, SwissFederal Institute of Technology, Zurich, 2008.

[27] S. Agarwal. Design guidelines for high efficiency photovoltaics and low power tran-sistors using quantum transport. PhD thesis, Purdue University, 2010.

[28] H. A. Atwater and A. Polman. Plasmonics for improved photovoltaic devices. NatureMaterials, 9:205–13, 2010.

17 Metamaterials

In this chapter we review the basic concept of metamaterials as those possessing simulta-neously negative permittivity and permeability over the same frequency range. Theoreticalprinciples and basic experimental results are reviewed. Possible applications includingcloaking, slow light and optical black holes are described.

17.1 Introduction

Metamaterials are artificially created structures with predefined electromagnetic properties.They are fabricated from identical elements (atoms) which form one-, two- or three-dimensional structures. They resemble natural solid state structures. Metamaterials typicallyform a periodic arrangement of artificial elements designed to achieve new propertiesusually not seen in Nature [1]. In a sense, they are composed of elements in the same wayas matter consists of atoms.

Metamaterials are characterized and defined by their response to electromagnetic wave.Optical properties of such materials are determined by an effective permittivity εeff andpermeability μeff valid on a length scale greater than the size of the constituent units.In order to introduce such a description, one requires that the size of artificial inclusionscharacterized by d be much smaller than wavelength λ, i.e. d � λ.

The name meta originates from Greek, μετα and means ‘beyond’. Main characteristicsof metamaterials (MM) are:

• man-made,• have properties not found in Nature,• have rationally designed properties,• are constructed by placing inclusions at desired locations.

With modern fabrication techniques it is possible to create structures which are muchsmaller than the wavelength of visible light. An example of the unusual properties that suchstructures could have is the negative refractive index. Although the problem has a longhistory (for example, Mandelstam [2] in 1945 discussed negative refraction and negativegroup velocity), it was only in 1967 when Victor Veselago [3] indicated the possibilityof materials having simultaneously negative μ and ε. He demonstrated the technologicalpotential that materials with a negative refractive index could have for applications inimaging. It recently made its way to optics, mostly due to rapid progress in nanofabrication.

384

385 Introduction

Fig. 17.1 Schematics of the elementary cells and the wavelength of external electromagnetic wave in two extreme cases.

The most well-known property of metamaterials is the negative index of refraction(NIM – negative index materials). Another popular name used is left-handed material. Thisterminology will soon be explained.

The situation is illustrated in Fig. 17.1. For a longer wavelength, one cannot sensethe properties of individual constituent atoms, shown on the left, whereas the shorterwavelength, on the order of the distance between ‘atoms’, can effectively be used todetermine some of the atom’s properties (like their locations). In metamaterials we dealwith the situation on the left.

Metamaterials can have controlled magnetic and electric responses over a broad range offrequencies. Those responses depend on the properties of individual elements (atoms). Inthe long-wavelength limit, where a � λ, where a is the characteristic dimension and λ is thewavelength of electromagnetic wave, one should perform some sort of averaging procedureto determine effective parameters of MM. Details of those procedures are discussed in theliterature [4]. In the end, it is possible to achieve the condition where εeff < 0 and μeff < 0.

A diagram which illustrates the classification of metamaterials is shown in Fig. 17.2. Ingeneral negative index materials do not exist in Nature; the rare exception is bismuth, whichwhen placed in a waveguide shows a negative refractive index at a wavelength of λ = 60µm[5]. There are no known naturally occurring NIMs in the optical range. However, artificiallydesigned materials (metamaterials) can act as NIM. One should, however, notice that theoccurrence of NIM needs both negative εeff < 0 and μeff < 0 over the same frequencyrange.

Metamaterials can open new avenues to achieve unprecedented physical properties andfunctionality unattainable with naturally existing materials. Optical NIMs promise to createentirely new prospects for controlling and manipulating light, optical sensing and nanoscaleimaging and photolithography.

17.1.1 Short history of MM

The earliest discussion of the concept of negative refraction goes probably to Schuster[6]. In the book entitled An Introduction to the Theory of Optics he noted that negativedispersion of the refractive index with respect to wavelength, i.e. dn/dλ < 0, can producenegative refraction of light entering such a medium from a vacuum. Some other early works

386 Metamaterials

conventional

(Permeability)

(Permittivity)LHM ferritessplit-ring structures

plasmawire structures

μ

ε

ε<0, μ<0 ε>0, μ<0

n<0

n>0

ε<0, μ>0 ε>0, μ>0

Fig. 17.2 Classification of materials according to the sign of electric permittivity and magnetic permeability.

on negative refraction include Lamb [7] published in 1904 and Pocklington [8] publishedin 1905. Historical aspects were summarized by Simovski and Tretyakov [9] and by Moroz[10].

In recent years the field of metamaterials has received remarkable attention with thenumber of published papers growing exponentially. This is due to unusual properties ofsuch systems (see [4] for a recent review) and also important practical applications likeperfect lenses [11], invisibility cloaking [12], [13], slow light [14] and enhanced opticalnonlinearities [15]. Parallel to theoretical developments, there has been noted spectacularexperimental progress in the field [16].

The subject of metamaterials is attracting enormous attention from researchers. Wesearched ISI Web of Knowledge, and as of October, 2011 there were 4545 published paperson the subject of ‘metamaterials’. The plot of the number of published papers is shown inFig. 17.3. As can be seen, the number of published papers grows exponentially with somesaturation observed recently.

The modern research progress started in 1999 with the pioneering work by Pendryand collaborators [17] who discussed artificial electromagnetic structures, such as split-ring resonators (SRR) (in general, two concentric split rings), for which they predicted theexistence of negative magnetic permeability. In those structures the incident electromagneticwave excites circulating currents in the loops. These currents in turn create oscillatingmagnetic dipole moments. With proper design, those currents are resonantly enhanced,leading to a negative magnetic permeability.

An SRR meta-atom can, in the long-wavelength (quasi-static) regime, be modelled asa resistor/inductor/capacitor (RLC) resonant circuit [18]. Stacks of such structures exhibitnegative permeability.

The creation of a negative electric permittivity is relatively easy. For instance, it is well-known that metals at optical frequencies exhibit a plasmon-like permittivity that can assume

387 Introduction

Year

Num

ber o

f pub

lishe

d pa

pers

1200

1000

800

600

400

200

0

2000 2002 2004 2006 2008 2010

Fig. 17.3 Number of papers published on the subject of ‘metamaterials’.

negative values. At smaller (e.g. microwave) frequencies, it was shown by Pendry et al.in 1996 [19] that an array of long wires acts as a diluted Drude metal when the electricfield is oriented along the wire axis. Such a system allows for the existence of a negativepermittivity below the effective plasma frequency. A combination of both magnetic andelectric elements in a suitable design leads to negative index of refraction over a specifiedfrequency band.

A report on the first negative-index material (NIM) designed and fabricated along theabove principles was published in 2000 by Smith et al. [20]. It operated at microwavefrequencies. The individual SRR had its resonance frequency at 4.845 GHz. The diameterof the wires was about 0.8 mm, which resulted in a plasma frequency of ωp = 13× 109s−1.There, the authors performed transmission measurements on a NIM structure having arange of frequencies over which the refractive index was negative for one direction ofpropagation. In a subsequent paper, Shelby et al. [21] (again in a transmission experiment)demonstrated a negative refraction of microwaves incident on the interface between air andNIM.

In 2003, a group from Boeing Phantom Works reported [22] a very accurate measure-ments of a Snell’s law using a NIM wedge at frequencies from 12.6 to 13.2 GHz.

Following those pioneering demonstrations, in the next few years the operation fre-quency has been increased by more than four orders of magnitude. In particular, Yenet al. [23] demonstrated a NIM response at 1 THz (λ = 300 µm) by making use ofSRRs placed on a double-sided polished quartz substrate. Further design and fabrica-tion improvements have led to the possibility of creating a three-dimensional double-negative medium with subwavelength meta-atoms in the optical regime [24], [25]. Indeed,in 2008, a three-dimensional photonic NIM at optical frequencies has been demonstrated inRef. [26].

388 Metamaterials

17.2 Veselago approach

In this section we summarize an early approach as developed by Veselago [3].

17.2.1 Wave equation

In order to proceed with a quantitative description, at this stage we will introduce Maxwell’sequations, which in the absence of free charges and currents are

∇ × E = − μ∂H

∂tand ∇ × H =ε

∂E

∂t(17.1)

Take ∇× (rotation) operation on the first equation and then use second equation

∇ × ∇ × E = − ε∂

∂t(∇ × H) = −με

∂2E

∂t2

Use general relation valid for arbitrary vector A

∇ × ∇ × A = ∇ (∇ · A) − ∇2A = −∇2A

In the last step, we used ∇ · E =0. One obtains the wave equation

∇2E = −με∂2E

∂t2= 0 (17.2)

If we disregard losses and consider ε and μ as real numbers, then one can observe thatwave equation is unchanged when we simultaneously change signs of ε and μ.

17.2.2 Left-handedmaterials

Start with the basic Maxwell’s equations (17.1). Assume time-harmonic fields

E(x, y, z, t) = E(x, y, z)eiωt + c.c.

and introduce wave vector k by assuming plane-wave dependence

E, H ∼ eik·r

Maxwell’s equations take the form

k × E = −μωH (17.3)

k × H = εωE (17.4)

From Eqs. (17.3) and (17.4) and definition of cross product, one can immediately seethat for ε > 0 and μ > 0 vectors E, H and k form a right-handed triplet of vectors, and ifε < 0 and μ < 0 they form a left-handed system, see Fig. 17.4.

389 How to create metamaterial?

E

H k E

H

k

(b)(a)

Fig. 17.4 (a) Right-hand orientation of vectors E, H, k for the case when ε > 0, μ > 0. (b) Left-hand orientation of vectorsE, H, k for the case when ε < 0, μ < 0. The figure is taken, with permission of the Canadian Association ofPhysicists (CAP), from an article by Wartak et al. [27].

1 2

θ

34

1

2

n1

n2

θ

Fig. 17.5 Reflection and refraction in negative index material. The figure is taken, with permission of the Canadian Associationof Physicists (CAP), from an article by Wartak et al. [27].

17.2.3 The refraction of a ray

Consider propagation of a ray through the boundary between left-handed and right-handedmedia, in Fig. 17.5. Light crossing the interface at non-normal incidence undergoesrefraction, that is a change in its direction of propagation. The angle of refraction dependson the absolute value of the refractive index of the medium. Here, 1 is the incident ray,2 is the reflected ray, 3 is the refracted ray assuming medium ′2′ is right-handed, and 4is the refracted ray when assuming that medium ′2′ is left-handed. In a metamaterial therefraction of light would be on the same side of the normal as the incident beam, seeFig. 17.5. The relation between angles is determined by Snell’s law.

17.3 How to createmetamaterial?

17.3.1 Metamaterials with negative effective permittivity in the microwave regime

At optical frequencies metals are characterized by an electric permittivity that varies withfrequency according to Drude relation

ε(ω) = ε0

[1 − ω2

p

ω(ω + iγ )

](17.5)

390 Metamaterials

E

B

k

a

z

yx

Fig. 17.6 Schematic illustration of a periodic arrangement of infinitely long thin wires along z direction used in the creation ofan effective plasma medium at microwave frequencies. The figure is taken, with permission of the CanadianAssociation of Physicists (CAP), from an article by Wartak et al. [27].

Here ω2p = Ne2

mε0is the plasma frequency, i.e. the frequency with which the plasma consisting

of free electrons oscillates in the presence of an external electric field. Typical values for ωp

are in the ultraviolet regime. The other symbols are: N is the electron density; e is charge ofan electron and m its mass. The parameter γ describes damping and its value, for examplefor copper is γ ≈ 4 × 1013 rad s−1.

In the limit when γ = 0, from Eq. (17.5) it follows that ε < 0 for ω < ωp; i.e. themedium is characterized by a negative permittivity. Considering typical values of ωp, theresulting range of negative values of ε is in the ultraviolet regime. Unfortunately, in thisfrequency range ω � γ , and as a result losses dominate the behaviour of ε. Thus, usingmetals to achieve negative ε over this frequency range will be impractical (high losses) andthe propagation of light will be mainly evanescent.

To achieve negative ε at microwave frequencies, Pendry et al. [28] proposed to useperiodic structure consisting of long thin metallic wires of radius r arranged on a horizontalplane (xy), see Fig. 17.6. The unit cell of this periodic structure is a square whose sideshave length equal to a.

When electric field E = E0e−i(ωt−kz)z is incident on this structure, it forces free electronsto move inside the wires in the direction of the incident field. Effective electron density Neff

of such structures which participate in plasma oscillations is Neff = N πr2

a2 (with N beingthe electron density inside each wire, r the radius of a wire and a distance between wires),which is significantly smaller compared to N , thus reducing the effective plasma frequency.For example, for a wire with radius r = 1 µm and wire spacing a =5 mm, one finds thatNeff ≈ 1.3 × 10−7N; i.e. the effective electron density of the new medium is reduced byseven orders of magnitude compared to that of the free electron gas inside an isolated wire.

Additionally, in such engineered structures the effective mass meff of the electrons issignificantly larger compared to that of a free electron. To determine effective mass of anelectron in this wired medium, we use the classical equation of motion of a moving electron:

d(mv)/dt = e[E + v × B] → d

dt[mv + eA] = −e∇(ϕ − v · A) (17.6)

where e is the charge of an electron and v the velocity of an electron. One also has B = ∇×Aand E = −∇ϕ − ∂A/∂t. We divide the xy plane into circles of radius Rc, centred at each


wire and having area equal to that of the square unit cell; i.e. Rc = a/√

π . Furthermore,we assume that the wires are sufficiently apart from each other so that the magnetic fieldinside each circle arises only from the current I that flows perpendicularly to the centre ofthe circle, and that the field at the circumference of each circle vanishes; i.e. H (Rc) = 0.Magnetic field intensity at a distance R from each wire is given by

H = I

2πR

(1 − R2

R2c

)(17.7)

Magnetic field H associated with a vector potential A according to the relation H =μ−1

0 ∇ × A gives the following expression for the vector potential A:

A = μ0I

2π

(ln(Rc/R) + R2 − R2

c

2R2c

)z (17.8)

where z is a unit vector along the z-direction. It has been assumed that A(R ≥ Rc) = 0. Fordistances very close to the wires, i.e. for R → r << Rc, Eq. (17.8) gives

A ≈ μ0I

2πln(a/r)z (17.9)

Since both v and A point along z, the right hand side of the second part of Eq. (17.6) willvanish (for a given R); i.e. the so-called ‘conjugate momentum’ (mv + eA) of an electronwill be conserved along the z-direction. As a result, an electron will be moving in the aboveengineered medium with an effective mass, meff = eA(r)/v. Realizing that the current I canbe simply re-expressed as I = −(πr2)(Nev), we finally obtain using Eq. (17.9) the effectivemass of a moving electron inside our effective medium meff = 0.5 × μ0Ne2r2ln(a/r).Thus, for copper wires of radius r = 1 µm, being separated by a =5 mm, we obtain: meff ≈1.3×104 m; i.e. the effective mass of an electron in our engineered medium is increased bymore than four orders of magnitude. This, combined with the fact that the effective electrondensity is reduced by approximately seven orders of magnitude, leads to an effective plasmafrequency that is in the microwave regime:

ω2p = Neff e2

meff ε0= 5.1 × 1010[rad s−1]2

→ νp = ωp/2π = 8.2 GHz (17.10)

It should be noted that, based on Eq. (17.10), the calculated wavelength, λp = c/νp, whichcorresponds to our medium’s effective plasma frequency, turns out to be considerably largercompared to the periodicity of the structure (λp ≈ 7a), justifying the description of theperiodic structure as an effective medium. Therefore we are able to construct an engineeredmedium that can exhibit a negative electric permittivity in the microwave regime withreasonably low losses.

17.3.2 Magnetic properties: split-ring resonators

In the previous section, we examined how to construct an artificial structure possessingnegative effective permittivity ε. Here, we will show how to create negative effectivepermeability μ in the microwave regime using the so-called split-ring resonators (SRR).

392 Metamaterials

Fig. 17.7 Perspective view of split-ring resonator.

x

z

y

d

r

I

a

a

lt

dc

x

z

y

I

B

)b()a(

(c)

Fig. 17.8 Arrangement of SRR in space. Reprinted with permission from H. Chen et al., J. Appl. Phys., 100, 024915 (2006).Copyright 2006, American Institute of Physics.

The perspective view of SRR is shown in Fig. 17.7. The structure consists of two cutcylinders. The model used to represent SRR is illustrated in Fig. 17.8. Here we will describedescription of split-ring resonators (SRR) based on equivalent circuit model approach [18].The structure under consideration is shown in Fig. 17.8.


I

Cg

Cg

L L'

M

U V

R

Fig. 17.9 Equivalent circuit of SRR.

We will establish the equivalent circuit model of this structure and show that it canproduce negative effective permittivity μ. The unit cell shown above can be modelled usingequivalent circuit model shown in Fig. 17.9.

Apply a time-varying external magnetic field H0 in the y-direction. It will induce acurrent I flowing in each SRR unit. From Faraday’s law, the voltage V due to the externalfield H0 is

V = iωμ0πr2H0 (17.11)

where r is the radius of the ring. Along the y-direction, SRR loops form a column (stack)which behaves like a solenoid. In each loop there is a current I flowing. Neglect the fringingeffects, i.e. spreading of the magnetic field lines. The magnetic flux in such a column isthus

� = πr2μ0I

l(17.12)

where l is the separation between loops. The inductance L appearing in the circuit model is

L = μ0πr2

l(17.13)

The inductance L is defined per ring of an infinite column of rings.Let L′ be the total inductance of the SRR units in the other column (excluding column

represented by inductance L). Coupling between L and L′ is represented by the mutualinductance M . For very long columns in y-direction, the mutual inductance M is

M = �L

I= lim

n→∞πr2

na2

φd

I= lim

n→∞πr2

na2(n − 1) L

= πr2

a2L = F · L (17.14)

Here �L is the flux of the depolarization field located in the interior of L, � = L · I is thedepolarization field generated by one column of the rings, φd = (n − 1) � = (n − 1) L · Iis the flux of the total depolarization field, and F = πr2

a2 is the fractional volume of theperiodic unit cell in the xz plane occupied by the interior of the SRR.

394 Metamaterials

Applying Kirchoff’s voltage drop law for a loop in the equivalent circuit shown inFig. 17.9 gives

V = R · I + I

−iωC+ (−iωL) I − (−iωM ) I (17.15)

where C = 12Cg is the total capacitance in the loop.

From the previous equations

V = iωμ0πr2H0 = iωL I H0 (17.16)

From Eq. (17.15) one has

V =(

R − 1

iωC− iωL + iωM

)I

Solving for current

I = V

R − 1iωC − iωL + iωM

= iω L I H0

R − 1iωC − iωL + iωF · L

(17.17)

= L H0R

iωL + 1ω2LC − 1 + F

= −L H0

(1 − F ) − 1ω2LC + i R

ωL

Magnetic dipole moment per unit volume of the material is

M = πr2

la2I

with the current I inferred from the above equation to be

I = − H0l

(1 − F ) − 1/(ω2LC) + iR/(ωL)(17.18)

As a result, the (relative) effective magnetic permeability associated with this medium willbe (in the direction, x, that the incident magnetic field is polarized)

μr = B/μ0

B/μ0 − Md= 1 − F

1 − 1/(ω2LC) + iR/(ωL)(17.19)

From Eq. (17.19) we can see that μ assumes negative values in the range: 1/√

LC <

ωp < 1/√

LC(1 − F ), where ωm0 = 1/√

LC is the resonance frequency of the Lorentzianvariation of the medium’s magnetic permeability, and ωmp = 1/

√LC(1 − F ) is the cor-

responding plasma frequency (where Re{μ} = 0). Crucially, we note that the resonantwavelength (λm0) of the structure depends entirely on the rings’ effective inductance (L)

and capacitance (C), and can therefore be made considerably larger than the periodicity (a)

of the structure, thereby fully justifying its description as an effective medium.Plot of real and imaginary parts of effective magnetic permeability is shown in Fig. 17.10

and the MATLAB code is in Appendix, Listing 17A.1.The combination of long wires and SRR in a unit cell is shown in Fig. 17.11. The effect

of both results in a negative effective permittivity and permeability over the same frequencyband.

We finish this section by showing in Fig. 17.12 schematic plots of effective permittivity(εeff ) and permeability (μeff ).

395 Some applications of metamaterials

0 2 4 6 8 10−20

−15

−10

−5

0

5

10

15

20

Frequency (GHz)

Eff

ectiv

e pe

rmea

bili

ty

Fig. 17.10 Real and imaginary parts of magnetic permeability.

I

E

B

z

yx

Fig. 17.11 Elementary cell of metamaterial formed by SRR and thin wire. The figure is taken, with permission of the CanadianAssociation of Physicists (CAP), from an article by Wartak et al. [27].

(a) (b)

mpm0p

Re{εeff}Re{μeff}

μeff =1

ω ω ω ω ω

Fig. 17.12 (a) Permittivity of wire medium demonstrating plasma-like frequency dependent permittivity and (b) Frequencyresponse of effective permeability. The figure is taken, with permission of the Canadian Association of Physicists (CAP),from an article by Wartak et al. [27].

17.4 Some applications of metamaterials

Materials with such unusual properties allow for unusual applications. We do not attemptto review all of them; we just concentrate on a few interesting possibilities. We start withthe possibility of creating a so-called ‘perfect lens’.

396 Metamaterials

source focus refocus

RH LH RH

Fig. 17.13 Double focussing of a source with a planar double negative (left-handed) metamaterials slab surrounded by regular(right-handed) dielectrics.

17.4.1 Perfect lenses

Until recently, it was thought that the manipulation of light is limited by the fundamentallaw of diffraction to a relatively long wavelengths (say around 0.5λ). The sub-wavelengthdetails are carried by evanescent harmonics which decay exponentially and are also subjectto noise. A conventional lens only collects the propagating waves. The evanescent wavesare lost due to their decay.

A planar slab consisting of NIM with sufficient thickness can act as a lens, knownas Veselago lens. Such a slab lens with refractive index n = −1 placed in vacuum (seeFig. 17.13) can resolve details of an object with subwavelength precision [11]. Detailedmathematical analysis is discussed, for example, by Ramakrishna and Grzegorczyk [4] andCai and Shalaev [29].

To understand the problem, consider a slab of thickness d and refractive index of, e.g.n = −1, surrounded by air. It will bring all rays emanating from a source to a double focus:first, at a point inside the NIM slab, at a distance s = l < d, where l is the distance of thesource from the slab, and second at a point outside the slab, at a distance d − l. Hence, sucha slab acts like a lens, and is able to bring the rays radiated by a source to a focus outsidethe slab, without reflections occurring at the media interfaces because the n = −1 slab isimpedance-matched to free space.

Evanescent waves are associated with the high spatial frequencies of the electromagneticwaves created by source. They carry fine (subwavelength) features of the source. Therefore,a NIM slab can, in principle, enable us to obtain the image of an object with ‘perfect’resolution, containing all the subwavelength features of an object and overcoming the usualdiffraction limitations that characterize conventional lenses.

17.4.2 Stopped light in metamaterials

For decades scientists maintained that optical data cannot be stored statically and must beprocessed and switched on the fly. The reason for this conclusion was that stopping andstoring an optical signal by dramatically reducing the speed of light itself was thought tobe unfeasible.

At present, some of the most successful slow-light designs based on photonic-crystals(PhCs) [30] or coupled-resonator optical waveguides (CROWs) [31], can indeed slow


a teff >aA B

C D

Fig. 17.14 Classical G-H shift in regular dielectrics. The figure is taken, with permission of the Canadian Association of Physicists(CAP), from an article by Wartak et al. [27].

a 0<teff<aA ABB

Fig. 17.15 G-H shift in NIM forming central layer. The figure is taken, with permission of the Canadian Association of Physicists(CAP), from an article by Wartak et al. [27].

down light efficiently by a factor of only 40; otherwise, large group-velocity-dispersionand attenuation-dispersion occur; i.e. the guided light pulses broaden and the attainablebandwidth is severely restricted.

Recently Tsakmakidis et al. [14] proposed a new method that can allow for a truestopping of light in NIM. The stopping of light in the proposed configuration is associatedwith a negative Goos-Hanchen (G-H) phase shift, a lateral displacement of light ray whenit is totally reflected at the interface of two different dielectric media. Classical G-H shiftbetween dielectrics having positive refractive indices is illustrated in Fig. 17.14.

In a structure where the central layer is formed by NIM, the G-H shift is reversed as it isillustrated in Fig. 17.15.

To more precisely understand the manner in which light is decelerated in this structure,let us imagine a ray of light propagating in a zig-zag fashion along a waveguide with anegative-index (‘left-handed’) core. The ray experiences negative Goos-Hanchen lateraldisplacements each time it strikes the interfaces of the core with the positive-index (‘right-handed’) claddings, see Fig. 17.15. Accordingly, the cross points of the incident and reflectedrays will sit inside the left-handed core and the effective thickness of the guide will besmaller than its natural thickness. It is reasonable to expect that by gradually reducingthe core physical thickness, the effective thickness of the guide will eventually vanish.Obviously, beyond that point the ray will not be able to propagate further down, and willeffectively be trapped inside the negative index metamaterial (NIM) heterostructure, seeFig. 17.16.

The authors of [14] proposed stopping a light pulse by varying the thickness of thewaveguide core to the point where the cycle-averaged power flow in the core and the

398 Metamaterials

ateff = 0A=B

Fig. 17.16 G-H shift in NIM at critical thickness. The figure is taken, with permission of the Canadian Association of Physicists(CAP), from an article by Wartak et al. [27].

cladding become comparable. At the degeneracy point, where the magnitudes of thesepowers become equal, the total time-averaged power flow directed along the central axisof the core vanishes. At this point the group (or energy) velocity goes to zero and thepath of the light ray forms a double light cone (‘optical clepsydra’) where the negativeGH lateral shift experienced by the ray is equal to its positive lateral displacement as ittravels across the core. Adiabatically reducing the thickness of the NIM core layer maythus, in principle, enable complete trapping of a range of light rays, each corresponding toa different frequency contained within a guided wavepacket.

This ability of metamaterial-based heterostructures to dramatically decelerate or evencompletely stop [32] light under realistic experimental conditions, has recently led to a seriesof experimental works [33], [34] that have reported an observation of so-called ‘trappedrainbow’ light-stopping in metamaterial waveguides.

17.4.3 Cloaking (invisibility)

Unusual light-bending properties constitute a rather generic feature of metamaterials. Partof the excitement surrounding these materials is that they could be engineered to ‘cloak’objects from electromagnetic radiation such as light: that is, make them seem invisible atspecific frequencies. A metamaterial ‘invisibility’ cloak can be designed such that it does notreflect waves back nor scatter them in other directions. Several methods were proposed tomake extended bodies invisible, such as those based on cancellation of scattering [35] or oncoordinate transformations. The method suggested by Pendry et al. [36] (see also Leonhardt[12]) relies on controlling the paths of electromagnetic waves. It was applied to a sphericalvolume and uses coordinate transformation that expels the paths of electromagnetic waves(rays) from a spherical volume, squeezing them into a spherical shell around the volumethat is to be cloaked, thereby making it invisible to incident radiation, see Fig. 17.17. Thelight rays smoothly avoid the cloaked object and flow around it like a fluid. They appearto have properties of the free space when observed externally. The rays make a detouraround the hidden part of the device. On the left side they must arrive at the same timeas if they were propagating through empty space. Since in the cloaking region they travellonger distances, their phase velocity must exceed c. This is in principle possible [37] (fora specific frequency).


Fig. 17.17 The rays go around the inner object and then go to the eye. An observer at the left of the cloak would see the pointsource.

To introduce the relevant transformation, one starts with a mathematical point (placing itat the origin of a coordinate system), which is obviously invisibly small. To hide an extendedobject, say a sphere of radius R1, we transform the mathematical point at the origin into asphere of radius R1 and the vicinity of the sphere into another sphere of radius R2 > R1.The transformation which does this is

r′ = R1 + R2 − R1

R2r, θ ′ = θ, φ′ = φ (17.20)

Maxwell’s equations are form-invariant to coordinate transformations like the one above.Only the components of μ and ε are affected by the transformation. They become spatiallyvarying and anisotropic tensors. Their forms were determined by Pendry et al. [36] and forR1 < r < R2 are

ε′r′ = μ′

r′ = R2

R2 − R1

(r′ − R1

)2

r′

ε′θ ′ = μ′

θ ′ = R2

R2 − R1(17.21)

ε′φ′ = μ′

φ′ = R2

R2 − R1

Inside the sphere with radius R1 the permittivity and permeability can be arbitrary, whereasthe space between the spheres is filled with a material having the permittivity and per-meability tensors determined by Eqs. (17.21). As a result, any object inside the spherewith radius r < R1 is concealed. These conclusions follow from exact manipulations ofMaxwell’s equations and are not restricted to a ray approximation.

17.4.4 Optical black holes

By extending the concept behind the invisibility cloak, one can speculate on the possibilityof creating an optical analogue of a black hole. In this concept, by proper design of anew class of metamaterials, one can expect to concentrate and trap light waves, similarlyto what can happen in a ‘black hole’, see Fig. 17.18. In such a system, light will bepermanently trapped. Such a black hole design has recently been proposed by Narimanovand Kildishev [38]. Numerical simulations showed a highly efficient light absorption. Theelectromagnetic black hole was built recently [39], [40] and it operates at microwave

400 Metamaterials

Fig. 17.18 Metamaterial black hole fabricated from properly crafted gradient-index material.

frequencies. The structure is composed of 60 concentric layers, and each layer is a thinprinted circuit board etched with a numbers of subwavelength unit structures on one side andcoated with 0.018 mm thick copper on the other side. The permittivity changes radially inthe shell of the microwave black hole, and hence the unit cells are identical in each layer buthave different sizes in adjacent layers. The structure can efficiently absorb electromagneticwaves coming from all directions owing to the local control of electromagnetic fields. It isexpected that such a microwave black hole could find important applications in solar-lightharvesting, thermal emitting, and cross-talk reduction in microwave circuits and devices.These specially designed analogues of black holes could also be used for controlling,slowing and trapping electromagnetic waves [40], as well as for investigating some of themore exotic physics associated with celestial mechanics.

17.5 Metamaterials with an active element

This last section is intended to briefly discuss the problem of losses in metamaterials andhow to compensate for them.

As mentioned before, metamaterials show large losses which at present are orders ofmagnitude too large for practical applications and are considered as an important factorlimiting practical applications of metamaterials. For example, detailed analytical studiesshow that losses limit the superresolution of a theoretical superlens [41]. There was somecontroversy whether loss elimination can be feasible [42], but as shown by Webb and Thylen[43] it is possible to completely eliminate losses in metamaterials.

Very recently, several computational [44], [45], [46] and experimental [47] works havedemonstrated that optical losses can be fully overcome in realistic negative-refractive-index metamaterials. The specific loss-free design considered in [44] and [47] consisted oftwo metallic films perforated with small rectangular holes (‘fishnets’), and with an activemedium (laser dye) spacer between two films. Additionally, several reports [48], [49],[50] speculated about possible compensation for losses in metamaterials by introducing again element. For example, Wegener et al. [49] formulated a simple model where gain isrepresented by a fermionic two-level system which is coupled via a local-field to a singlebosonic resonance representing the plasmonic resonance of the metamaterial. Also recently,

401 Appendix 17A

Fang et al. [50] described a model where the gain system is modelled by a generic four-level atomic system. They conducted numerical analysis using the FDTD technique. Gainmaterial was introduced in the gap region of the split-ring resonators (SRR). The systemhad a magnetic resonance frequency at 100 THz. Some other reports of the design andanalysis of active metamaterials are by Yuan et al. [51] and Sivan et al. [32].

17.6 Annotated bibliography

We finish this short review by highlighting some recently published textbooks where furtheruseful information about metamaterials can be found. We start with the most recent:

• T. J. Cui, D. R. Smith, R. Liu, eds., Metamaterials. Thory, Design, and Applications,Springer, 2010.

Concentrates on the recent progress in metamaterials, in particular the optical trans-formation theory, invisible cloaks, new type of antennas and optical metamaterials.

• W. Cai and V. Shalaev, Optical Metamaterials. Fundamentals and Applications, Springer,2010.

Details recent advances on optical metamaterials from fundamental aspects to currentimplementations.

• F. Capolino, ed., Metamaterials Handbook. Vol. 1: Theory and Phenomena of Metama-terials, Vol. 2: Applications of Metamaterials, CRC Press, Boca Raton, 2009.

A very broad collection of topics covered by many authors.• L. Solymar and E. Shamonina, Waves in Metamaterials, Oxford University Press, 2009.

Offers a comprehensive treatment of all aspects of research in this field at a levelthat should appeal to final year undergraduates in physics or in electrical and electronicengineering.

• S. A. Ramakrishna and T. M. Grzegorczyk, Physics and Applications of Negative Refrac-tive Index Materials, SPIE and CRC Press, 2009.

Discussion of principles of negative refraction and comparison with other media thatexhibit similar properties.


In Table 17.1 we provide a list of MATLAB files (in fact, only one file) created for Chapter 17and a short description of its function.



17A.1 ChenFig3.m Reproduces Fig. 3 from the paper by Chen et al. [18]

402 Metamaterials

Listing 17A.1 Program ChenFig3.m.

% File name: ChenFig3.m

% Matlab code to plot Fig.3 of Chen et al

% using equivalent circuits approach

% Plot Fig3. of Chen et al, JAP,v.100, 024915 (2006)

clear all

% Fundamental constants

epsilon_zero=8.8542d-12; % epsilon zero (F/m)

mu_zero = 4*pi*10^-7; % mu zero (H/m)

%

% Geometrical dimensions, see Fig.1 of Chen

r = 2.0; % internal radius = 2 mmm

a = 5.0; % distance between elements = 5 mm

l_vert = 1.0; % vertical separation = 1 mm

L_c = 1.0; % Dimension of capacitor = 1 mm

t = 0.8; % Dimension of capacitor = 0.8 mm

d_c = 0.2; % Dimension of capacitor = 0.2 mm

eps_r = 4.0; % permittivity of the gap

sigma = 10^-6; % resistance per unit length

%

% Intermediate parameters

F = pi*r^2/(a^2); % fractional volume (dimensionless)

C_g = (epsilon_zero*eps_r*L_c*t/d_c)*10^-3; % gap capacitance unit (F)

L = (mu_zero*pi*r^2/l_vert)*10^-3; % Eg.(2) unit (H)

% pause

R = 2*pi*r*sigma; % resistance of the ring

C = C_g/2.0; % total capacitance in the loop

%

% Determination of frequency range


% in the interval [0,10] GHz

freq_GHz = linspace(0,10,N_max); % creation of frequency arguments

freq_Hz = freq_GHz*10^9;

omega_Hz = 2.0*pi*freq_Hz;

%

tt = 1./(omega_Hz.^2*L*C);

ttt = R./(omega_Hz.*L);

temp = 1./tt + 1i*ttt;

%

% effective permeability without coupling M, Eq.(9) of Chen

extra = F./(1. + F - tt + 1i*ttt);

%

mu_eff = 1. - F./(1. + F - tt + 1i*ttt);

result_re = real(mu_eff);

result_im = imag(mu_eff);

403 References

%

% plot(freq_GHz,result_im);

% pause

plot(freq_GHz,result_re,freq_GHz,result_im,’.’,’LineWidth’,1.5);

axis([0 10 -20 20])

xlabel(’Frequency (GHz)’,’FontSize’,14);

ylabel(’Effective permeability’,’FontSize’,14);


grid

pause

close all

References

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[2] L. I. Mandelstam. Group velocity in crystal lattice. Zh. Eksp. Teor. Fiz., 15:475–8,1945.

[3] V. G. Veselago. The electrodynamics of substances with simultaneously negativevalues of ε and μ. Soviet Physics Uspekhi, 10:509–14, 1968.

[4] S. A. Ramakrishna and T. M. Grzegorczyk. Physics and Applications of NegativeRefractive Index Materials. SPIE Press and CRC Press, Bellingham, WA, 2009.

[5] V. A. Podolskiy, L. Alekseev, and E. E. Narimanov. J. Mod. Optics, 52:2343–9, 2005.[6] A. Schuster. An Introduction to the Theory of Optics. Edward Arnold, London, 1904.[7] H. Lamb. On group-velocity. Proc. London Math. Soc, 1:473–9, 1904.[8] H. C. Pocklington. Nature, 71:607–8, 1905.[9] C. R. Simovski and S. A. Tretyakov. Historical notes on metamaterials. In F. Capolino,

ed., Theory and Phenomena of Metamaterials, pages 1: 1–17. CRC Press and Taylorand Francis Group, Boca Raton and London, 2009.

[10] A. Moroz, www.wave-scattering.com/negative.html, 3 July 2012.[11] J. B. Pendry. Negative refraction makes a perfect lens. Phys. Rev. Lett., 85:3966–9,

2000.[12] U. Leonhardt. Optical conformal mapping. Science, 312:1777–80, 2006.[13] D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and

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[16] V. M. Shalaev. Optical negative-index metamaterials. Nature Photonics, 1:41–8, 2007.

404 Metamaterials

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[18] H. Chen, L. Ran, J. Huangfu, T. M. Grzegorczyk, and J. A. Kong. Equivalent circuitmodel for left-handed metamaterials. J. Appl. Phys., 100:024915, 2006.

[19] J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs. Extremely low frequencyplasmons in metallic mesostructures. Phys. Rev. Lett., 76:4773–6, 1996.

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[21] R. A. Shelby, D. R. Smith, and S. Schultz. Experimental verification of a negativeindex of refraction. Science, 292:77–9, 2001.

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[23] T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, andX. Zhang. Terahertz magnetic response from artificial materials. Science, 303:1494–6,2004.

[24] N. Liu, H. Guo, L. Fu, S. Kaiser, H. Schweizer, and H. Giessen. Three-dimensionalphotonic metamaterials at optical frequencies. Nature Materials, 7:31–7, 2008.

[25] J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang. Opticalnegative refraction in bulk metamaterials of nanowires. Science, 321:930, 2008.

[26] J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, andX. Zhang. Three-dimensional optical metamaterial with a negative refractive index.Nature, 455:376–80, 2008.

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[28] J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart. Low frequency plasmonsin thin-wire structures. J. Phys. Condens. Matter, 10:4785–809, 1998.

[29] W. Cai and V. Shalaev, eds. Optical Metamaterials. Fundamentals and Applications.Springer, New York, 2010.

[30] B. Corcoran, C. Monat, C. Grillet, D. J. Moss, T. P. Eggleton, B. J. White, L. O.O’Faolain, and T. F. Krauss. Green light emission in silicon through slow-light en-hanced third-harmonic generation in photonic crystal waveguides. Nature Photonics,3:206–10, 2009.

[31] F. Xia, L. Sekaric, and Y. Vlasov. Ultracompact optical buffers on a silicon chip.Nature Photonics, 1:65–71, 2006.

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[33] X. P. Zhao, W. Luo, J. X. Huang, Q. H. Fu, K. Song, C. Cheng, and C. R. Luo.Trapped rainbow effect in visible light left-handed heterostructures. Appl. Phys. Lett.,95:071111, 2009.

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[36] J. B. Pendry, D. Schuring, and D. R. Smith. Controlling electromagnetic fields. Science,312:1780, 2006.

[37] P. W. Milonni. Fast Light, Slow Light and Left-Handed Light. Institute of PhysicsPublishing, Bristol and Philadelphia, 2005.

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[42] M. I. Stockman. Criterion for negative refraction with low optical losses from afundamental principle of causality. Phys. Rev. Lett., 98:177404, 2007.

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[45] K. L. Tsakmakidis, M. S. Wartak, J. J. H. Cook, J. M. Hamm, and O. Hess. Negative-permeability electromagnetically induced transparent and magnetically-active meta-materials. Phys. Rev. B, 81:195128, 2010.

[46] A. Fang, Th. Koschny, and C. M. Soukoulis. Self-consistent calculations of loss-compensated fishnet metamaterial. Phys. Rev. B, 82:121102, 2010.

[47] S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H. K. Yuan, and V. M.Shalaev. Loss-free and active optical negative-index metamaterials. Nature, 466:735–40, 2010.

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A Appendix A Basic MATLAB

In this Appendix we discuss basic elements of MATLAB, including code design andspecific examples of programming in MATLAB. Some introductory books are: MATLABProgramming for Engineers by Chapman [1], MATLAB. An Introduction with Applicationsby Gilat [2] and Mastering MATLAB 5. A Comprehensive Tutorial and Reference byHanselman and Littlefield [3] – a reference book with a broad range of examples.

MATLAB can operate in two modes:

1. from the command window (interactive mode), where one introduces commands atMATLAB prompt

2. from the m-file (script). All commands are written in a file (with *.m extension). Suchfiles are activated (run) by typing their names (without an extension) at MATLABprompt. Each *.m file can call several other *.m files.

We would always advocate using m-files. They can be reused, combined with otherprograms, etc. In short, all code can be used in both ways.

The Appendix is divided into several sections, each dealing with different aspects ofMATLAB. In the code provided, we put many comments describing its workings. Wesuggest that in each case the reader runs the code and analyses its outputs.

In Table A.1 we provide a list of MATLAB files created for this Appendix A and a shortdescription of each function.

Before we start a more detailed description of MATLAB and its rules, we will introduceand run sessions dealing with fundamental MATLAB issues.

Table A.1 List of MATLAB functions for Chapter A.


A.1 intro session.m Preliminary use of MATLABA.2 memory.m Compares execution times for various memory handlingA.3 loops.m Compares execution times for various loopsA.4 basic 2D plot.m Basic 2D MATLAB plotA.5 sub plots.m Introduces subplotsA.6 pview.m 3D plotA.7 f ile write.m Writes to a fileA.8 f ile read.m Reads from fileA.9 deriv.m Numerical differentiation using MATLAB function di f f ()

406

407 Working session with m-files

A.1 Working session withm-files

Here, we will illustrate preliminary use of MATLAB using the m-file. The full code namedintrosession.m is shown below. We suggest run this code and observe the results. Try alsoto modify it and analyse the outputs. We put many comments inside the code to explain itsoperation.

% File name: intro_session.m

%----------------------- Initialization --------------------------

clear all

’its me’ % outputs text on screen

pause

d=6; % assigns value 6 to variable d

C=[1 2 3; 3 2 1; 4 5 6];% defines matrix C

C

who % lists defined matrices

pause

whos % lists the matrices and their sizes

pause

%------------------ Colon operator -------------------------------

x =C( : , 3) % selects third column of C matrix

pause

format long % shows variables on a screen in a long format

% format short

t = 0:0.1:2;

t % lists value of t

pause

t’ % transpose operator, creates vertical vector

pause

%------------- Special values and special matrices -----------------

pi % shows the value of pi

1i

Inf

clock % year, month, day, hour, minute, seconds

pause

date % date in string format

eps % the smallest number on my computer

pause

A = zeros(4) % creates 4x4 matrix consisting of zeros

B = zeros(4,3) % creates 3x4 matrix consisting of zeros

pause

A1 = ones(3) % creates 3x3 matrix consisting of ones

B1 = ones(4,3) % creates 3x4 matrix consisting of ones

408 Basic MATLAB

G = ones(1,5) % creates vector, length 5 consisting of ones

pause

D = eye(3) % creates identity matrix

z=input(’input z’);

z % shows inputed number

pause

%------------- Prints a short table of cos values ----------------

clc % clears screen

n = 21; x = linspace(0,1,n); y = cos(2*pi*x);

disp(’ ’)

disp(’k x(k) cos(x(k))’)

disp(’------------------------’)

for k=1:n

degrees = (k-1)*360/(n-1);

fprintf(’ %2.0f %5.0f %8.3f \n’,k,degrees,y(k));

end

disp( ’ ’);

disp(’x(k) is given in degrees.’)

pause

%------------- Output Options ------------------------------------

a = 1.23456789;

format short

a % display a in short format

pause

format long

a % display a in long format

disp(a) % another display of a in long format

fprintf(’Its me,\n %4.24f ’, a)

pause

%--------------------- Data files ----------------------------

save data_1 x y; % as *.mat file

pause

load data_1;

z=x’

save data_5.dat z /ascii;

pause

%------------- Scalar and array operations -----------------------

clear all

2+5

A=[1 2 3; 3 2 1; 4 5 6]; B=[2 3 4; 4 5 6; 5 6 7];

%

C1=A+B % addition of two matrices

C2=A*B % matrix multiplication

409 Basic rules

C3=A.*B % element by element multiplication

C4=A./B % element by element division

C5=A.^B % element by element exponentiation

pause

A.2 Basic rules

Several of the special symbols used in MATLAB are listed in Table A.2. All variablesin MATLAB are treated as matrices. Each operation is therefore considered as a matrixoperation and as such has been optimized for vector and matrix operations. For the bestuse, one should vectorize algorithms and loops and also reserve in advance memory for allobjects (matrices). We summarize basic rules and illustrate them with preliminary examples.

• vectorization of loops. This means replacement of for and while loops by proper vectoror matrix notation.

Suppose that we need to generate 101 values of sin function in the interval from 0 to 1.

It can be done in two ways:1. using f or loop (no vectorization)

t0 = clock; for n = 0:0.001:1

x = sin(n)

end

time_no_vect = etime(clock,t0)

Run time on my computer is time no vect = 0.2300.

2. applying vectorization as shown below

t0 = clock;

n = 0:0.001:1;

Table A.2 Some of the special symbols used in MATLAB.

Symbol Description

= assigning value[] construction of an array+ array addition− array subtraction.∗ array multiplication∗ matrix multiplication./ array right division′ transpose operator% beginning of a comment

410 Basic MATLAB

x = sin(n);

time_vect = etime(clock,t0);

Now, run time on the same machine is time vect = 0.0810. In order to determinetime difference between start and end points, we used MATLAB function etime, whichaccurately computes time differences over many orders of magnitude.

• reserve memory.• use colon (:) notation• limit using f or in loops• use m-files with functions instead of scripts• provide extensive comments• create descriptive names for variables• use dot (.) notation

Some of the above rules will be discussed now in more detail.

A.3 Some rules about good programming in MATLAB

Here we summarize several general rules of programming including those specific toMATLAB (but not solely).

1. Always comment your program, so another person (even yourself after several months)can understand it.• use meaningful variable names,• use units consistently in your formulas or (better) work with dimensionless (or scaled)

variables.2. All calculations in MATLAB can be either performed in the Command Window or m-

files can be created containing the same source code as typed in the Command Window.m-files play the role of functions or subroutines.

3. A semicolon at the end of all MATLAB statements suppresses printing the assignedvalues on the screen, in the Command Window. This speeds up program execution.

A.3.1 Preallocate memory

Always preallocate memory for the generated matrix. It increases execution speed becauseMATLAB does not need to increase matrix size after determining each element. Runand experiment with program memory res.m shown below. A significant difference in theexecution time can be observed.

% File name: memoryMSW.m

% comparison of execution time with and without memory reservation

clear all

% regular loop without memory allocation

411 Some rules about good programming in MATLAB

% we use functions tic and toc to measure elapsed time

N = 10000;

tic; % start timer

n = 0;

for x = 1:N

n = n + 1;

y(n) = cos(2*pi*x);

end


%

% the same loop with preallocated memory

tic

n = 0;

y =zeros(1,N);

for x = 1:N

n = n + 1;

y(n) = cos(2*pi*x);

end

toc

A.3.2 Vectorize loops

MATLAB has been designed for vectorial operations. They execute much faster than regularloops. The comparison of both methods is illustrated in the program loops.m shown below.We use MATLAB functions tic and toc to measure elapsed time.

% File name: loops.m

% comparison of execution time for regular loop and vector loop

clear all

% regular loop is executed first

% we use functions tic and toc to measure elapsed time

tic; % start timer

n = 0;

for x = 1:0.1:1000

n = n + 1;

y(n) = cos(2*pi*x);

end


%

% the same loop is vectorized below

tic

x = 0:0.1:1000;

y = cos(2*pi*x);

toc

412 Basic MATLAB

A.4 Basic graphics

Very good quality graphics is one of the reasons for MATLAB’s popularity. Here wesummarize the basic rules.

A.4.1 Basic 2D plot

A basic plot is produced in two steps:1. Creating two arrays of equal lengths, one containing values of independent variable x

and the other containing dependent variable y = f (x).Creation of an array of values of independent variable can be done in several ways. Below

we created arrays of points ranging from 0 to 2π using different methods.

x1 = 0:pi/100:2*pi; % 100 evenly spaced points

x2 = [0:0.25:2*pi]; % predefined spacing of 0.25

x3 = linspace(0,2*pi,100); % 100 evenly spaced points

2. Inputting created arrays x and y into the MATLAB plot function.The steps are illustrated in the MATLAB program below.

clear all % clears all variables in memory

x = 0:pi/100:2*pi; % creates array containing independent variable

y = cos(2*x); % creates array containing function values

plot(x,y) % creates generic plot, without description

pause % pause, to view the plot

close all % closes all figures

A.4.2 Two-dimensional plots

The main function used here is plot. It can be called up with different number of arguments

plot(y) plot(x,y plot(x,y,’line_type’)

Here, we illustrate several ways to use it.Listing: Basic 2D plot.Here we show creation of the simplest two-dimensional plot by plotting function y = 3x,

adding basic description and grid. The graph generated by basic 2D plot.m file is shownin Fig. A.1.

MATLAB code is shown below. It explains creation of a basic two-dimensional graph.

% File name: basic_2D_plot.m

% Basic plotting of MATLAB are explained.

clear all

% Generate data for plot

n = 100; % number of plotting points

x = linspace(0,10,n); % generates points on x-axis

413 Basic graphics

0 2 4 6 8 100

5

10

15

20

25

30Plot of function 3*x

x value

y va

lue

Fig. A.1 Basic two-dimensional plot.

y = 3*x; % function to be plotted

% Creation of basic plot and adding description

h = plot(x,y); % basic plot

pause % stop to analyse plot

title(’Plot of function 3*x’,’FontSize’,14) % adding title

xlabel(’x value’,’FontSize’,14);% adding text on x-axis and size of x label

ylabel(’y value’,’FontSize’,14);% adding text on y-axis and size of y label

%

pause(2) % stop for 2 seconds

grid % adding grid



%

pause % final stop

close all % closing all windows

Listing: Creating subplots.Function subplot (2,2,1) divides the Figure window into 2 vertical and 2 horizontal

windows and places the current plot into window 1 (top, left). In the remaining code othertypes of plots are created and placed in the remaining subwindows.

The results of sub plots.m are shown in Fig. A.2. MATLAB code is shown below. Itexplains the creation of subplots.

% File name: sub_plots.m

% Creation of subplots with different properties.

clear all

% Generate data for plot

414 Basic MATLAB

0 5 100

100

200

300

10−1 100 1010

100

200

300

0 5 1010−5

100

105

10−1 100 10110−5

100

105

Fig. A.2 Creation of subplots with different properties.

n = 100; % number of plotting points

x = linspace(0,10,n); % generates points on x-axis

y = 3*x.^2; % function to be plotted

subplot(2,2,1) % division of Figure window

h1 = plot(x,y); % basic plot shown in top-left window


pause % stop to contemplate effects

% The above function is now plotted on different log scales

% and plots are placed in the remaining sub-windows

subplot(2,2,2)

h2 = semilogx(x,y); % log scale on x-axis


subplot(2,2,3)

h3 = semilogy(x,y); % log scale on y-axis

set(gca,’FontSize’,15); subplot(2,2,4)

h4 = loglog(x,y); % log scale on both axes


%

% Below we set new thicknesses of plotting lines

set(h1,’LineWidth’,1.5); set(h2,’LineWidth’,2);

set(h3,’LineWidth’,2.5); set(h4,’LineWidth’,3.5);

pause

close all

Types of lines (both colours and symbols) are summarized in Table A.3. Colours areautomatically selected by MATLAB.

415 Basic graphics

Table A.3 Colours and symbols used to create lines.

Symbol Colour Symbol Type of line

y yellow . pointm magenta o circlec cyan x shown symbolr red + shown symbolg green * shown symbolb blue - continuousw white : shown symbolk black -. dash-dotted line

0

5

10

0

0.2

0.4

0.6

0.8

1−1

0

1

xy

z

Fig. A.3 A 3D plot.

A.4.3 Some 3D plots

We show how to put several 2D plots on one graph and make it look like a 3D plot. Thecode is shown below and the generated graph is as Fig. A.3.

% File name: pview.m

% Allows multiple 2D plots to be stacked next to one another

% along one dimension; also provides 3D view of all plots

clear all

x = linspace(0,3*pi).’; % x-axis data

Z = [sin(x) sin(2*x) sin(3*x) sin(4*x)];

% Code below gives each curve different value on y-axis

Y = [zeros(size(x)) ones(size(x))/3 (2/3)*ones(size(x)) ones(size(x))];

plot3(x,Y,Z,’LineWidth’,1.5)

grid on

xlabel(’x’,’FontSize’,14)

416 Basic MATLAB

ylabel(’y’,’FontSize’,14)

zlabel(’z’,’FontSize’,14)


view(-40,60)

pause

close all

A.5 Basic input-output

A.5.1 Writing to a text file

To save the results of some calculations to a file in text format requires the following steps:

a) Open a new file, or overwrite an old file, keeping a ‘handle’ for the file.b) Print the values of expressions to the file, using the file handle.c) Close the file, using the file handle.

The file handle is just a variable which identifies the open file in your program. Thisallows you to have any number of files open at any one time.

% File name: file_write.m

% open file

fid = fopen(’myfile.txt’,’wt’); % ’wt’ means "write text"

if (fid < 0)

error(’could not open file "myfile.txt"’);

end;

for i=1:10 % write to file

fprintf(fid,’Number = %3d Square = %6d\n’,i,i*i);

end;

fclose(fid); % close the file

A.5.2 Reading from a text file

To read some results from a text file is straightforward if you just want to load the wholefile into the memory. This requires the following steps:

a) Open an existing file, keeping a ‘handle’ for the file.b) Read expressions from the file into a single array, using the file handle.c) Close the file, using the file handle.

The fscanf() function is the inverse of fprintf(). However, it returns the values it reads asvalues in a matrix. You can control the ‘shape’ of the output matrix with a third argument.

% File name: file_read.m

% open file

417 Numerical differentiation

fid = fopen(’myfile.txt’,’rt’); % ’rt’ means "read text"

if (fid < 0)

error(’could not open file "myfile.txt"’);

end;

% read from file into table with 2 rows and 1 column per line

out = fscanf(fid,’Number = %d Square = %d\n’,[2,inf]);

fclose(fid); % close the file

xx = out’; % convert to 2 columns and 1 row per line.

xx % output results to screen

A.6 Numerical differentiation

We will illustrate numerical differentiation using MATLAB function diff(x). It computesthe difference between elements in an array.

Given a vector x = {...xk−1, xk, xk+1...} containing points xk , one can derive approxima-tions to the first derivative f ′(x) = dy

dx of the function y = f (x). Three basic possibilitiesare [4]

Backward difference, f ′(xk ) ≈ f (xk ) − f (xk−1)

xk − xk−1(A.1)

Forward difference, f ′(xk ) ≈ f (xk+1) − f (xk )

xk+1 − xk(A.2)

Central difference, f ′(xk ) ≈ f (xk+1) − f (xk−1)

xk+1 − xk−1(A.3)

To compute first derivative we use MATLAB function diff(x), which determines thedifference between adjacent values of the vector x. The derivative is then determined as

dy = diff(y)./diff(x);

The returned array of differences contains one less element than the original array. Oneshould observe that trimming the last value of x produces a forward difference and trimmingthe first value gives a backward difference.

To obtain the second derivative, we apply the above algorithm a second time. Usingbackward difference, one has

f ′′(xk ) ≈ f ′(xk ) − f ′(xk−1)

xk − xk−1(A.4)

% File name: deriv.m

% Program evaluates first and second derivatives and plots results

clear all

font_size = 18;

N_max = 190;

x = linspace(0,2,N_max);

418 Basic MATLAB

y1 = sin(pi.*x);

h1 = plot(x,y1); % plot of original function

xlabel(’x’,’FontSize’,font_size);

ylabel(’Original function’,’FontSize’,font_size);

grid on

pause

%

temp1 = y1;

dy1 = diff(temp1)./diff(x);

xnew1 = x(1:length(x)-1);

h2 = plot(xnew1,dy1); % plots first derivative


ylabel(’First derivative’,’FontSize’,font_size);

grid on

pause

%

temp2 = dy1;

dy2 = diff(temp2)./diff(xnew1);

xnew2 = xnew1(1:length(xnew1)-1);

h3 = plot(xnew2,dy2); % plots second derivative


ylabel(’Second derivative’,’FontSize’,font_size);

grid on

pause

close all

A.7 Review questions

1. How are variables represented in MATLAB?2. What is the role of the semicolon (;) operator in MATLAB?3. What does MATLAB function linspace(0, 1, 10)?4. What is the role of dot (.) operator in MATLAB?5. What does MATLAB function subplot(3, 2, 1)?6. Perform a polar plot.7. Estimate π using the random method.

References

[1] S. J. Chapman. MATLAB Programming for Engineers. Brooks/Cole, Pacific Grove,CA, 2000.

419 References

[2] A. Gilat. MATLAB. An Introduction with Applications. John Wiley & Sons, Hoboken,NJ, 2008.

[3] D. Hanselman and B. Littlefield. Mastering MATLAB 5. A Comprehensive Tutorial andReference. Prentice Hall, Upper Saddle River, NJ, 1998.

[4] S. E. Koonin. Computational Physics. The Benjamin/Cummings, Menlo Park,CA, 1986.

B Appendix B Summary of basic numerical methods

In a book like this, the development of computer programs for various tasks and alsoexecution of simulations for different processes and devices, plays an essential role. Thefundamentals of many computer programs are supported by numerical methods. Therefore,in this Appendix we summarize main elements of numerical analysis with an emphasis onmethods related to the development of programs used in this book, and also to understandingof operation of those programs.

There are many excellent textbooks devoted to numerical analysis. We found the booksby Koonin [1], DeVries [2], Garcia [3], Gerald and Wheatley [4], Rao [5], Heath [6] andRecktenwald [7] of significant pedagogical value. The books by Press et al. [8] stand ontheir own as an excellent source of practical computer codes ready to use.

We concentrate on description and implementation of some practical numerical methodsand not on the problems which those methods are typically used for. We start our discussionwith a summary of methods of solving nonlinear equations.

There are many textbooks aimed to the introduction of numerical methods and their ap-plications. Some of the most popular are: Applied Numerical Analysis Using MATLAB byFausett [9], Numerical Methods for Physics by Garcia [3], Introduction to Scientific Com-puting by van Loan [10], Advanced Engineering Mathematics with MATLAB by Harmanet al. [11], A Friendly Introduction to Numerical Analysis by Bradie [12].

They contain extensive code written in MATLAB (sometimes also in other languages)which should help with understanding and could be easily adopted to a particular problem.

Our aim here is to summarize some of the numerical methods and techniques which aredirectly relevant to the problems discussed in the main text.

In Table B.1 we provide a list of MATLAB files created for this Appendix B and a shortdescription of each function.

B.1 One-variable Newton’s method

The very efficient and popular method, also known as Newton-Raphson method, is used tofind roots. Here we establish its principles by showing how to find root x∗ for an arbitraryfunction of single variable f (x) such that

f (x∗) = 0 (B.1)

420

421 One-variable Newton’s method

Table B.1 List of MATLAB functions for Appendix B.


B.1 muller.m Function implements Muller’s methodB.2 dmuller.m Driver to test Muller’s methodB.3 f muller.m Test functions for Muller’s methodB.4 ode single.m Driver for RK method (single eq.)B.5 f unc ode.m Function called by ode single.mB.6 f unc ode sys.m Function called by ode sys.mB.7 ode sys.m Driver for RK method (system of eqs.)B.8 square wave.m Implements Fourier series for square waveB.9 FT example 1.m Preliminary Fourier transformB.10 FT example 2.m Testing Fourier transform in MATLAB

x

f(x)

x1x2x*

Fig. B.1 Graphical illustration of Newton’s method.

The Newton method is based on a linear approximation of the function by employing atangent to this function at a particular point. In Fig. B.1 we provide graphical illustrationof Newton’s method.

One starts from an initial guess point, say x1, which should be not too far from root x∗.Evaluating tangent at point x1 and its intersection with the x axis creates new point x2 andthe process is repeated. The relevant mathematical formulas are obtained by writing theTaylor expansion of f (x) around initial guess x1:

f (x∗) = f (x1 − δx) = f (x1) − df (x1)

dxδx + O(δx2) (B.2)

In the previous, f (x∗) = 0. Solving for δx, one finds

δx = f (x1)

f ′(x1)

From Fig. B.1 one observes that point x2 can be obtained as

x2 = x1 − δx = x1 − f (x1)

f ′(x1)(B.3)

The procedure can be generalized and for arbitrary step n, one has

xn+1 = xn − f (xn)

f ′(xn), n = 1, 2, . . . (B.4)

422 Summary of basic numerical methods

B.2 Muller’s method

Muller’s method is an efficient technique of finding both real and complex roots of ascalar-valued function f (x) of a single variable x where there is no information aboutits derivatives. It uses quadratic interpolation involving three points [13]. The method isdescribed in several textbooks, including Atkinson [14] and Fausett [9].

In order to illustrate how the method works, let us start with three initial guesses x0, x1, x2.By applying Muller’s method, a new improved approximate point x3 is determined. In thismethod one uses quadratic function y(x) which is used in fitting original function f (x).Function y(x) is of the form

y(x) = a (x − x2)2 + b (x − x2) + c (B.5)

It contains three constants a, b, c which can be determined by evaluating Eq. (B.5) at pointsx0, x1, x2. One obtains

for x = x2 f (x2) = c

for x = x1 f (x1) = a (x1 − x2)2 + b (x1 − x2) + c

or f (x1) − f (x2) = a (x1 − x2)2 + b (x1 − x2)

for x = x0 f (x0) = a (x0 − x2)2 + b (x0 − x2) + c

or f (x0) − f (x2) = a (x0 − x2)2 + b (x0 − x2)

Introduce notation u0 = x0 − x2 and u1 = x1 − x2, which allows us to write the aboveequations as

f (x0) − f (x2) = a u20 + b u0 (B.6)

f (x1) − f (x2) = a u21 + b u1 (B.7)

The system of Eqs. (B.6) and (B.7) can be solved by any standard method and one obtains

a = u1[

f (x0) − f (x2)

]− u0[

f (x1) − f (x2)

](x0 − x2) (x1 − x2) (x0 − x1)

b = u20

[f (x1) − f (x2)

]− u21

[f (x0) − f (x2)

](x0 − x2) (x1 − x2) (x0 − x1)

With the above information at hand and using Eq. (B.5), one can determine roots of

y(x) = 0

in order to generate a new approximation of the root of f (x). One obtains

x − x2 = −b ± √b2 − 4ac

2a(B.8)

Out of the above two roots we choose the one which is closest to x2. To avoid round-off errorsdue to subtraction of nearly equal numbers, one multiplies numerator and denominator of

423 Muller’s method

(B.8) by −b ∓ √b2 − 4ac and obtains (we call new solution x by x3)

x3 − x2 =(−b ± √

b2 − 4ac) (

−b ∓ √b2 − 4ac

)2a(−b ∓ √

b2 − 4ac)

= − 2c

−b ± √b2 − 4ac

(B.9)

Once x3 is determined, the ‘oldest’ point x0 is ignored by setting x0 = x1, x1 = x2, x2 = x3

and the process is repeated. MATLAB implementation of Muller’s method using the abovealgorithm is shown below.

% File name: muller.m

function out = muller(f, x0, x1, x2 , epsilon, max)

% Finds zeroes using Muller’s method, good for complex roots.


% out - result of search

% x1,x2,x3 - previous guesses

% epsilon - tolerance

% max - max number of iterations

%

y0 = f(x0);

y1 = f(x1);

y2 = f(x2);

iter = 0;

while (iter <= max)

iter = iter + 1;

a =( (x1 - x2)*(y0 - y2) - (x0 - x2)*(y1 - y2)) / ...

( (x0 - x2)*(x1 - x2)*(x0 - x1) );

%

b = ( ( x0 - x2 )^2 *( y1 - y2 ) - ( x1 - x2 )^2 *( y0 - y2 )) / ...

( (x0 - x2)*(x1 - x2)*(x0 - x1) );

%

c = y2;

%

if (a ~= 0)

disc = b*b - 4*a*c;

q1 = b + sqrt(disc);

q2 = b - sqrt(disc);

if (abs(q1) < abs(q2))

dx = - 2*c/q2;

else

dx = - 2*c/q1;

end

elseif (b ~= 0)

dx = - c/b;

end


x3 = x2 + dx;

x0 = x1;

x1 = x2;

x2 = x3;

%

y0 = y1;

y1 = y2;

y2 = f(x2);

%

if (abs(dx) < epsilon)

out = x2; break;

end

end

Driver for the Muller method is shown below:

% File name: dmuller.m

% Driver to test Muller’s method

clear all

format short

max = 100;

epsilon = 1e-6;

% starting points

x1 = 0.5;

x2 = 1.5;

x3 = 1.0;

% call to Muller’s method

out = muller(@fmuller, x1, x2, x3 , epsilon, max)

Test functions for the Muller method are shown below. One can make appropriate choices.

% File name: fmuller.m

function out = fmuller(x)

% Test functions for Muller’s method

out = x.^6 - 2;

%out = x.^10 - 0.5;

%out = x - x.^3/3;

%out = x.^3 - 5*x.^2 + 4*x;

B.2.1 Tests of Muller’s method

Using the above functions, we tested Muller’s algorithm on two functions:

1. calculations of 6√

2, after Fausett [9]. We define function f (x) = x6 − 2, and assumestarting values to be: x0 = 0.5, x1 = 1.5, x2 = 1.0. Obtained solution is x = 1.1225.


2. finding root of f (x) = ex + 1 = 0. The analytical method gives x = iπ . Assumingstarting values x0 = 1, x1 = 0, x2 = −1, one obtains from Muller’s method x =−0.0000 + 3.1416i.

B.3 Numerical differentiation

Very often scientific and engineering problems are formulated in terms of differentialequations. To solve them numerically, we need to establish schemes to replace derivativesby finite differences.

A typical scientific or engineering problem is described by the second-order differentialequation. Examples include Newton’s second law m

..x = F , or Laplace’s equation V ′′(x) =

0. General second-order differential equation is written as

d2y(x)

dx2+ a(x)

dy(x)

dx= b(x)

with a(x) and b(x) known functions. The above can be rewritten as two first-order equations:

dy(x)

dx= z(x)

dz(x)

dx= b(x) − a(x)z(x)

where z(x) is a new variable. The process can be generalized for a differential equationof arbitrary order, say n − th order, and when applied, results in n first-order differentialequations, as

dyi(x)

dx= fi(x, y1, y2, . . . yn), i = 1, 2, . . . n

where the functions fi on the right-hand side are known.In order to have a well-defined mathematical problem, the above set of equations must

be supplemented by boundary (or initial) conditions. Boundary conditions are algebraicconditions on the value of the functions yi to be satisfied at discrete specified points, butnot between those points.

The (first) derivative of a function f (x) is defined as

df (x)

dx

∣∣∣∣x0

= f ′(x0) = limx→x0

f (x) − f (x0)

x − x0

For a finite (and nonzero) value of �x = x−x0, the above derivative can be approximatedas

f ′(x0) ≈ f (x) − f (x0)

�x

In Table B.2 we summarize common finite-difference formulas.


Table B.2 Summary of common finite-difference formulas.

Type of approximation Formula Truncation error

Forward differences f ′i = fi+1 − fi

hO(h)

f ′′i = fi+2 − 2 fi+1 + fi

h2

Backward differences f ′i = fi − fi−1

hO(h)

f ′′i = fi − 2 fi−1 + fi−2

h2

Central differences f ′i = fi+1 − fi−1

2hO(h2)

f ′′i = fi+1 − 2 fi + fi−1

h2

xxi xi+1xi-1 xi+2xi-2

fi fi+1fi-1 fi+2fi-2

f(x)

Fig. B.2 Illustration used in determining derivatives.

B.3.1 Numerical differentiation using Taylor’s series expansion

We want to determine a derivative of function f (x) at point xi and also its second derivative.Using notation from Fig. B.2, we write Taylor expansions

fi+1 = fi + h f ′i + 1

2h2 f ′′

i + 1

6h3 f ′′′

i + O(h4) (B.10)

fi−1 = fi − h f ′i + 1

2h2 f ′′

i − 1

6h3 f ′′′

i + O(h4) (B.11)

where the following notation has been used:

fn = f (xn), xn = n · h, n = 0,±1,±2, . . .

All numerical schemes defining the first derivative are derived from the above equations.From Eq. (B.10) we obtain the forward formula

f ′i = fi+1 − fi

h+ O(h)

From Eq. (B.11) we obtain the backward formula

f ′i = fi − fi−1

h+ O(h)


xxixi+1xi-1

fi

f i+1

fi-1

f(x)

h h

Fig. B.3 Illustration of interpolation process.

Subtracting both, we obtain the central formula

f ′i = fi+1 − fi−1

2h

Taylor expansion for next points (i ± 2) gives

fi+2 = fi + 2h f ′i + 1

2(2h)2 f ′′

i + 1

6(2h)3 f ′′′

i + O(h4) (B.12)

and

fi−2 = fi − 2h f ′i + 1

2(2h)2 f ′′

i − 1

6(2h)3 f ′′′

i + O(h4) (B.13)

Multiplying Eq. (B.10) by 2 and adding to Eq. (B.12), we obtain

fi+2 − 2 fi+1 = fi + 2h f ′i + 2h2 f ′′

i − 2 fi − 2h f ′i − h2 f ′′

i + O(h3)

= − fi + h2 f ′′i + O(h3)

or

f ′′i = fi+2 − 2 fi+1 + fi

h2

Other schemes can be obtained in a similar way.

B.3.2 Numerical differentiation using interpolating polynomials

To illustrate a basic concept, consider a second-order polynomial f (x) passing throughthree data points, Fig. B.3:

(xi, fi) , (xi+1, fi+1) , (xi+2, fi+2)

The analytical form of that polynomial is

f (x) = a0 + a1x + a2x2 (B.14)


Function f (x) has to pass through all three points defined previously. This requirementcreates three equations:

fi = a0 + a1xi + a2x2i

fi+1 = a0 + a1(xi + h) + a2(xi + h)2

fi+2 = a0 + a1(xi + 2h) + a2(xi + 2h)2

The above equations will now be solved to find coefficients a0, a1, a2. The scheme mustbe independent of the choice of point xi, so we can choose xi = 0, which significantlysimplifies algebra. One obtains

fi = a0

fi+1 = a0 + a1h + a2h2

fi+2 = a0 + 2a1h + 4a2h2

The solutions are

a0 = fi

a1 = − fi+2 + 4 fi+1 − 3 fi

2h(B.15)

a2 = fi+2 − 2 fi+1 + fi

2h2

Differentiate Eq. (B.14) and use Eqs. (B.15) to obtain

f ′i = f ′(xi = 0) = a1 = − fi+2 + 4 fi+1 − 3 fi

2h

f ′′i = f ′′(xi = 0) = 2a2 = fi+2 − 2 fi+1 + fi

h2

The above are forward-difference approximations of the order O(h2).Given a vector x = {. . . xk−1, xk, xk+1 . . .} containing points xk , one can derive approxi-

mations to the first derivative f ′(x) = dydx of the function y = f (x). Three basic possibilities

are [1]:

Backward difference, f ′(xk ) ≈ f (xk ) − f (xk−1)

xk − xk−1(B.16)

Forward difference, f ′(xk ) ≈ f (xk+1) − f (xk )

xk+1 − xk(B.17)

Central difference, f ′(xk ) ≈ f (xk+1) − f (xk−1)

xk+1 − xk−1(B.18)

To compute first derivative we use MATLAB function diff(x), which determines thedifference between adjacent values of the vector x. The returned array of differencescontains one less element than the original array. One should observe that trimming thelast value of x produces a forward difference and trimming the first value gives a backwarddifference.


To obtain the second derivative, we apply the above algorithm a second time. Using thebackward difference, one has

f ′′(xk ) ≈ f ′(xk ) − f ′(xk−1)

xk − xk−1(B.19)

B.3.3 Crank-Nicolson method

We illustrate the method on the diffusion equation, which we write in the form

∂u

∂t= ∂2u

∂x2(B.20)

The Crank-Nicolson (CN) method is the finite-difference discretization of the above equa-tion. It was invented to overcome stability limitations and to improve the rate of convergenceof the solution of diffusion equation. It has O (�t)2 convergence rate compared to O (�t)of the implicit and explicit methods.

The CN method is essentially an average of the implicit and explicit methods. It iscreated as follows. Applying a forward difference approximation to the time derivative inthe diffusion equation gives the explicit scheme

un+1i − un

i

�t+ O (�t) = un

i+1 − 2uni + un

i−1

(�x)2+ O (�x)2 (B.21)

When using backward difference to the same equation, one creates the implicit scheme

un+1i − un

i

�t+ O (�t) = un+1

i+1 − 2un+1i + un+1

i−1

(�x)2+ O (�x)2 (B.22)

The algebraic average of the two schemes gives

un+1i − un

i

�t+ O (�t) = 1

2

{un

i+1 − 2uni + un

i−1

(�x)2+ un+1

i+1 − 2un+1i + un+1

i−1

(�x)2

}+ O (�x)2

(B.23)From the above equation one finally obtains the Crank-Nicolson scheme

un+1i − 1

2α(un+1

i+1 − 2un+1i + un+1

i−1

) = uni + 1

2α(un

i+1 − 2uni + un

i−1

)(B.24)

where α = �t(�x)2 . One can show (see Problems) that the scheme is accurate to O (�t)2

rather than O (�t). Note that in CN scheme un+1i+1 , un+1

i , un+1i−1 are determined implicitly in

terms of uni+1, un

i , uni−1.

B.3.4 Simple methods of numerical differentiation

Euler method

The simplest method is the Euler method, which is

yn+1 = yn + h f (xn, yn)


which advances a solution from xn to xn+1 = xn + h. It is not a practical method, because itis not very accurate and also not very stable (see discussion below).

Let us illustrate several methods which originate from the Euler method. We will usedefinition of an acceleration in one-dimensional motion as an illustration. It is

d2x(t)

dt2= a(x, v)

In general, acceleration can depend on position and velocity. First, write the above equationas a system of two first-order differential equations:

dv

dt= a(x, v) (B.25)

dx

dt= v

Introduce τ as a time step. The right-derivative formula for a general function g(t) is

dg(t)

dt= g(t + τ ) − g(t)

τ+ O(τ )

Application of that formula to a system (B.25) gives

v(t + τ ) − v(t)

τ+ O(τ ) = a(x(t), v(t))

x(t + τ ) − x(t)

τ+ O(τ ) = v(t)

We have explicitly indicated the dependence of all quantities involved on the specific time.From the above

v(t + τ ) = v(t) + τ · a(x(t), v(t)) + O(τ 2)

x(t + τ ) = x(t) + τ · v(t) + O(τ 2)

For the future use, let us introduce the following notation:

gn ≡ g((n − 1)τ ), n = 1, 2, 3, . . . (B.26)

and

g1 ≡ g(t = 0)

In the above, n refers to time steps. In our notation, the Euler method for the accelerationproblem reads

vn+1 = vn + τ · an

xn+1 = xn + τ · vn

The algorithm based on Euler method goes as follows:

1. specify initial conditions (i.e. values corresponding to t = 0) x1 and v1

2. choose time step τ

3. calculate acceleration for current values of xn and vn


4. use Euler method to compute new x and v5. loop through step 3, until all time steps are done.

An implementation of the Euler method is left as a problem.

Euler-Cromer (E-C) andmid-point methods

Both of these methods are simple modifications of the Euler method just discussed. E-Cmethod consists in replacing vn by vn+1, the value of velocity at the next time step (it mustbe calculated first, so the calculations must be done in a proper sequence). Explicitly, theE-C method is

vn+1 = vn + τ · an

xn+1 = xn + τ · vn+1

Another possible modification (and in a similar spirit) results in a mid-point method:

vn+1 = vn + τ · an

xn+1 = xn + τ · 1

2(vn + vn+1)

where the last term represents algebraic average of velocities at present and the next timesteps.

Leap-frog method

The development of this method starts with Eqs. (B.25). Use the central derivative approachto write (note that central time value is different for both equations)

v(t + τ ) − v(t − τ )

2τ+ O(τ 2) = a(x(t))

x(t + 2τ ) − x(t)

2τ+ O(τ 2) = v(t + τ )

Applying notation (B.26), the above system reads

vn+1 − vn−1

2τ= a(xn)

xn+2 − xn

2τ= vn+1

or

vn+1 = vn−1 + 2τ · a(xn)

xn+2 = xn + 2τ · vn+1

In order that this scheme works, we need to know values at proper initial points. Notethat position and velocity are evaluated at different points, as

position is evaluated at points: x1, x3, x5, . . .

velocity is evaluated at points: v2, v4, v6, . . .


Verlet method

To develop the Verlet method, instead of using Eqs. (B.25), we will write them as [3]

dx

dt= v

d2x

dt2= a

Use central difference formulas for first and second derivatives and obtain

xn+1 − xn−1

2τ+ O(τ 2) = vn

xn+1 + xn−1 − 2xn

τ 2+ O(τ 2) = an

From above, we obtain formulas which define Verlet method:

vn = 1

2τ(xn+1 − xn−1)

xn+1 = 2xn − xn−1 + τ 2an

B.4 Runge-Kutta (RK) methods

B.4.1 Second-order Runge-Kutta

Our goal is to solve linear system of linear differential equations

d−→x (t)

dt= −→

f (−→x (t), t)

with vector −→x expressed as −→x = [x1x2 . . . xn]. In order to establish second-order RKmethod, let us start with the simple Euler approximation to the first derivative (which isalso Taylor expansion to first order):

−→x (t + τ ) = −→x (t) + τ · −→f (

−→x (ξ ), ξ )

where τ is some arbitrary time step and t < ξ < t + τ. We have freedom to choose ξ

to have the most convenient value for us. The second-order RK method originates whenξ = t + τ/2 and also

−→x (t + τ ) = −→x (t) + τ · −→f (

−→x ∗(t + τ/2), t + τ/2)

where

−→x ∗(t + τ/2) = −→x (t) + 1/2 · τ · −→f (

−→x (t), t)

433 Solving differential equations

B.4.2 Fourth-order Runge-Kutta

This is one of the best schemes available and is commonly used in practice. It is defined as

−→x (t + τ ) = −→x (t) + 1/6 · τ · (−→F 1 + 2

−→F 2 + 2

−→F 3 + −→

F 4)

where−→F 1 = −→

f (−→x , t)

−→F 2 = −→

f (−→x + τ/2 · −→

F 1, t + τ/2)

−→F 3 = −→

f (−→x + τ/2 · −→

F 2, t + τ/2)

−→F 4 = −→

f (−→x + τ · −→

F 3, t + τ )

MATLAB has some built-in routines implementing RK methods.

B.5 Solving differential equations

Here we provide some practical methods of solving ordinary differential equations (ODE),both single and systems, using MATLAB function ode45.m.

B.5.1 Single differential equation

To start with, consider an equation with an initial condition

dx(t)

dt= x(t)

(2

t− 1

), x(0) = 0.009 (B.27)

We want to solve it and plot the solution in the interval [0.1, 10]. MATLAB code is dividedinto two m-files: one (the driver) contains all needed parameters and calls the appropriatem-function which defines the function.

% File name: ode_single.m

% Illustrates application of R-K method to a single diff. equation

% Results are compared with an exact solution

%

clear all

tspan = [0.1 10]; % time interval

x0 = 0.009; % initial value

%

[t,x] = ode45(’func_ode_single’,tspan,x0);

t_ex = linspace(0.1, 10, 100);

x_ex = t_ex.^2.*exp(-t_ex); % Exact solution

plot(t,x,t_ex,x_ex,’.’,’LineWidth’,1.5)


0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

t

x

Fig. B.4 Comparison of the solution of a single ordinary differential equation using numerical and analytical methods.

xlabel(’t’,’FontSize’,14)

ylabel(’x’,’FontSize’,14)


pause

close all

The function itself is

function xdot = func_ode_single(t,x)

% Function called by ode_single.m

xdot = x*(2/t -1);

The result of solving the above problem is plotted in Fig. B.4 along with an exact solution,which is x(t) = t2e−t .

B.5.2 System of differential equations

To illustrate application of ode45.m function in solving systems of differential equations,consider the Lorenz model [3] defined by the following system:

dx1(t)

dt= σ (x2 − x1)

dx2(t)

dt= rx1 − x2 − x1x3 (B.28)

dx3(t)

dt= x1x2 − bx3

where σ, r, b are positive constants. This system was originally developed to describebuoyant convection in a fluid. It shows chaotic behaviour. The following initial conditionsare assumed: x1(t) = 1, x2(t) = 1, x3(t) = 10.04. The analysis is conducted in the timeinterval [0, 10]. That system of equations is coded as

435 Numerical integration

function xdot = func_ode_sys(t,x)

% Function called by ode_sys.m

%

sigma = 10; b = 7/3; r = 25; % parameters

%

xdot(1) = sigma*(x(2) - x(1));

xdot(2) = r*x(1) - x(2) - x(1)*x(3);

xdot(3) = x(1)*x(2) - b*x(3);

xdot = xdot’;

The MATLAB code of the driver of the above system is

% File name: ode_sys.m

% Illustrates application of R-K method to system of diff. equations

%

clear all

tspan = [0 10]; % time interval

x0 = [1, 1, 10.04]; % initial value

%

[t,x] = ode45(’func_ode_sys’,tspan,x0);

plot(t,x(:,1), t,x(:,2), t,x(:,3),’LineWidth’,1.5)

xlabel(’time’,’FontSize’,14)


pause

close all

The results of two runs using the above functions are shown in Fig. B.5.

B.6 Numerical integration

The problem is to calculate a numerically definite integral

I =∫ b

af (x)dx

The interval [a, b] can be split into equidistant subintervals with total number N = b−ah ,

with h separation between neighbours. The above integral is thus replaced by

I =∫ b

af (x)dx =

∫ xi

af (x)dx +

∫ x2

xi

f (x)dx + · · · =∑

i

Ai

where Ai is the area of an elementary element. In the following, we therefore only consideran integral within a single subinterval:

Ai =∫ xi

xi−1

f (x)dx (B.29)

Different schemes will originate depending on how we evaluate the above integral.


0 2 4 6 8 10−20

−10

0

10

20

30

40

time

0 2 4 6 8 10−20

−10

0

10

20

30

40

time

Fig. B.5 Solution of a system of ordinary differential equations showing chaos for two different values of initial conditions;[1,1,10.04] (top) and [1,1,10.02] (bottom).

B.6.1 Euler’s rule

Subarea is defined within an interval xi−1 < xi < xi+1. The integral (B.29) thus takes theform

Ai =∫ xi

xi−1

f (x)dx ≈ h · fi

with fi = f (xi) taken at the right end of subinterval. The value of the function can also betaken at the left point, or in fact at any point within the subinterval xi−1 < xi < xi+1. Totalarea is thus

I ≈∑

i

Ai

= h · ( f1 + f2 + · · · fn)

437 Numerical integration

B.6.2 Trapezoidal rule

In this method the integral is approximated by a series of trapezoids. The area of eachsubinterval is evaluated as

Ai =∫ xi

xi−1

f (x)dx ≈ 1

2( fi−1 + fi) · (xi − xi−1)

with h = xi − xi−1. Taking h as a constant, the integral is

I =∫ b

af (x)dx ≈

∑i

1

2( fi−1 + fi) · h

= 1

2h {( f0 + f1) + ( f1 + f2) + ( f2 + f3) + · · · + ( fn−2 + fn−1) + ( fn−1 + fn)}

= 1

2h { f0 + 2 f1 + 2 f2 + · · · 2 fn−1 + fn}

In a condensed form, the integral is

I = hn−1∑i=1

fi + 1

2h ( f0 + fn)

B.6.3 Simpson’s rule

Simpson’s method gives more accurate results than trapezoidal rule. Trapezoidal rule con-nects consecutive points by a straight line; Simpson’s rule connects three points. Normally,a second-order polynomial (a parabola) is used to do so. Consider one:

p(x) = c2x2 + c1x + c0

Without loss of generality, we can take xi = 0. Then, xi−1 = −h and xi+1 = h. Evaluatingthe above polynomial gives

p(−h) = c2h2 − c1h + c0 = fi−1

p(0) = c0 = fi

p(h) = c2h2 + c1h + c0 = fi+1

Solving the above system gives

c0 = fi

c1 = 1

2h( fi+1 − fi−1)

c2 = 1

2h2( fi+1 + fi−1 − 2 fi)


Evaluating elementary integral for an area Ai gives

Ai =∫ h

−hp(x)dx

=∫ h

−h

{1

2h2( fi+1 + fi−1 − 2 fi) x2 + 1

2h( fi+1 − fi−1) x + fi

}dx

= 1

3h ( fi−1 + 4 fi + fi+1)

The complete integral is

I =∫ b

af (x)dx =

∑i

Ai

= 1

3h { f0 + 4 f1 + 2 f2 + 4 f3 + · · · + 2 fn−2 + 4 fn−1 + fn}

B.7 Symbolic integration in MATLAB

MATLAB allows us to perform symbolic manipulations. It uses Maple, a powerful com-puter algebra system, to manipulate and solve symbolic expressions. To use symbolicmathematics, we must use the syms operator to tell MATLAB that we are using a symbolicvariable, and that it does not have a specific value.

In this book we only use symbolic manipulations to evaluate indefinite integral. Weillustrate symbolic concepts of MATLAB with the example below.

Example Evaluate integral of the function f (x) = x2.The shortest method is to type on a prompt: >> int(x2). The answer is ans = x3/3.Alternatively, we can define x symbolically first, and then remove the single quotes in

the int statement.

>> syms x

>> int(x^2)

ans =

x^3/3

B.8 Fourier series

According to Fourier, any function can be expressed in the form known as the Fourierseries:

f (t) = 1

2a0 +

∞∑n=1

an cos nt +∞∑

n=1

bn sin nt (B.30)

439 Fourier series

Fourier series representation is useful in describing functions over a limited region, say[0, T ], or on the infinite interval (−∞,+∞) if the function is periodic. We assume thatthe Fourier series converges for any problem of interest, see Arfken [15]. Cosine and sinefunctions form a complete set and are orthonormal on any 2π interval. Therefore they maybe used to describe any function. The orthogonality relations are∫ 2π

0sin mt sin nt dt =

{πδm,n, m �= 0

0, m = 0(B.31)

∫ 2π

0cos mt cos nt dt =

{πδm,n, m �= 0

2π, m = n = 0(B.32)

∫ 2π

0sin mt cos nt dt = 0, all integral m and n (B.33)

One also has the relation1

2π

∫ 2π

0

(eimt)∗

eint dt = δm,n (B.34)

Using the above orthogonality relations, the coefficients of expansion in Eq. (B.30) are

an = 1

π

∫ 2π

0f (t) cos nt dt (B.35)

and

bn = 1

π

∫ 2π

0f (t) sin nt dt (B.36)

Also, the exponential form of the Fourier series is often used where the function is expandedas

f (t) =n=+∞∑n=−∞

cn eint (B.37)

Comparison of the above relations gives formulas for coefficients cn as

c0 = 12 a0 n = 0

cn = 12 (an − ibn) n > 0

c−n = 12

(a|n| + ib|n|

)n < 0

(B.38)

B.8.1 Change of interval

In the above formulation, the discussion has been restricted to an interval [0, 2π ]. If thefunction f (t) is periodic with the period 2T , we may write

f (x) = 1

2a0 +

∞∑n=1

an cos nt +∞∑

n=1

bn sin nt (B.39)

where

an = 1

T

∫ T

−Tf (t ′) cos

nπt ′

Tdt ′, n = 0, 1, 2, 3, . . . (B.40)


1

-1

T 2T t

f (t)

Fig. B.6 Square wave.

and

bn = 1

T

∫ T

−Tf (t ′) sin

nπt ′

Tdt ′, n = 1, 2, 3, . . . (B.41)

Expressions (B.30) and (B.39) are related by obvious change of variables, namely t = πt ′T .

The choice of the symmetric interval [−T, T ] is not essential; in fact, for periodic functionwith a period 2T it is possible to choose any interval [t0, t0 + 2T ].

B.8.2 Example

Use the Fourier series to approximate a square wave function f (t) as shown in Fig. B.6.Using Heaviside step function H (t), the function f (t) can be written as

f (t) = 2

[H

(t

T

)− H

(t

T− 1

)]− 1

where

H (t) ={

0 t < 0

1 t > 0

Determine coefficients an and bn in the Fourier series expansion. Write a MATLAB programwhich plots the above approximation for various number of terms in the series.

From Fig. B.6 one observes that f (t) = f (2T − t); i.e. function f (t) is odd. Thereforecoefficients an vanish. Coefficients bn are evaluated as follows:

bn = 1

T

∫ 2T

0f (t) sin

(nπt

T

)dt = 1

T

∫ T

0sin

(nπt

T

)dt − 1

T

∫ 2T

Tsin

(nπt

T

)dt =

= 1

nπ

∫ nπ

0sin xdx − 1

nπ

∫ 2nπ

nπ

sin xdx

={

4nπ

n odd

0 n even

The Fourier series is

f (t) = 4

π

∞∑n=1,3,5,...

1

nsin

(nπt

T

)The MATLAB program which evaluates the above is shown overleaf. The result is shownin Fig. B.7.

441 Fourier transform

−0.2 0 0.2 0.4 0.6 0.8 1

−1

−0.5

0

0.5

1

f(t)

t

Fig. B.7 Fourier series approximation to the square wave. Line with oscillations corresponds toN = 20 terms; the other onewas obtained using 100 terms.

% File name: square_wave.m

clear all

L = 0.5;

t = [0:100]/100; % arguments of independent variable

hold on

for N = [20 100];

f = zeros(1,101); % function initialization

for n = 1:2:N

pref = 4/(pi*n); % evaluate prefactor

f = f + pref*sin(n*pi*t/L); % summation of function

end

plot(t,f,’LineWidth’,1.5)

end

axis([-0.2 1.2 -1.2 1.2])

ylabel(’f(t)’,’FontSize’,14)

xlabel(’t’,’FontSize’,14)


pause

close all

B.9 Fourier transform

Based on the results from previous section, we now introduce Fourier transform (FT).Consider function f (t) to be nonperiodic on the infinite interval. FT is introduced byconsidering first a Fourier series representation (B.30) of the function which is periodic onthe interval [0, T ], where T is the period. First, make substitution

t → π

Tt


in Eq. (B.37) and obtain

f (t) =n=+∞∑n=−∞

cn ein πtT , cn = 1

T

∫f (t) e−in πt

T dt (B.42)

Next, introduce discrete angular frequencies

ωn = nπ

T

The difference between two successive frequencies �ω is evaluated as

�ω = ωn+1 − ωn = (n + 1)π

T− n

π

T= π

T

Using the above definition, the original series given by Eq. (B.42) can be written as

f (t) =n=+∞∑n=−∞

cn eint�ω, cn = 1

T

∫ +T

−Tf (t) e−int�ωdt (B.43)

Introducing new function g by the relation

cn = �ω√2π

g (n · �ω) (B.44)

one obtains

g (n�ω) = 1√2π

∫ +T

−Tf (t) e−int�ωdt (B.45)

and

f (t) = 1√2π

n=+∞∑n=−∞

�ωg (n�ω) eint�ω (B.46)

Taking the limit T → ∞, summation becomes an integral and n�ω becomes continuousvariable ω. One finds

g(ω) = 1√2π

∫ +∞

−∞f (t) e−iωtdt (B.47)

and

f (t) = 1√2π

∫ +∞

−∞g(ω) eiωtdt (B.48)

The above relations are known as the Fourier transform and inverse Fourier transform.There is a freedom in choosing prefactor in Eq. (B.44). By making different choices, onecan introduce different Fourier transforms.

443 FFT in MATLAB

B.10 FFT in MATLAB

Fast Fourier transform (FFT) in MATLAB is defined as

X (k) =N∑

n=1

x(n)e− j2π(k−1)(n−1)/N , 1 ≤ k ≤ N (B.49)

which corresponds to the analytical definition (B.47) but without 1√2π

factor. FFT in MAT-LAB is performed by the function fft.m.

The inverse FFT (IFFT) which is performed by MATLAB function ifft.m is defined inMATLAB as

x(n) = 1

N

N∑k=1

X (k)e j2π(k−1)(n−1)/N , 1 ≤ k ≤ N (B.50)

and it corresponds to the analytical definition (B.48). Again, the 1√2π

factor is absent. Letus illustrate a simple application of FT in MATLAB.

Example 1 As the simplest illustration of fft.m and ifft.m functions, we provide belowMATLAB code to conduct FT and its inverse for a Gaussian pulse. In the end, one recoversoriginal Gaussian pulse.

% File name: FT_example_1.m

% Doing preliminary Fourier transform in Matlab

clear all

N = 320;

T_0 = 10;

T_domain = 200;

Delta_t = T_domain/N;

T = Delta_t*(-N/2:1:(N/2)-1);

A_in = exp(-(T/T_0).^2); % input gaussian pulse

plot(A_in) % plot of input gaussian pulse

pause

A_freq = fft(A_in);

plot(A_freq) % not a correct pulse

pause % (I explain it in the next example)

orig = ifft(A_freq);

plot(orig) % plots original input gaussian pulse

pause

close all

Example 2 As the previous example shows, Fourier transform in MATLAB must be donecarefully. In this example we illustrate the workings of FFT in MATLAB. MATLAB code


0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1

Number of points

y(x)

y = exp(−x2)

−20 −10 0 10 20−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8F = fft(y)

Fig. B.8 Original Gaussian pulse (left) and and its FT (right).

is shown in the following. First, a Gaussian pulse is created and plotted against a numberof points along the x-axis. When needed, one can also plot it against x values. At the top ofeach figure (as a title) we provide an expression for the plotted function.

As the second figure, we plotted normalized Fourier transform using MATLAB functionf f t(). The plot does not resemble the expected result because the Fourier transformproduces a complex function of the frequency. Vector F contains complex numbers andwhen you plot it in MATLAB, it plots the real part against the imaginary part.

To correct for the above deficiency, in the third figure we plotted the above transformplus the MATLAB abs() function. Altogether, both functions produce the desired result,i.e. magnitude of the Fourier transform of the Gaussian pulse.

In the remaining two figures we performed the inverse Fourier transform of the createdFourier transform.

Observe the use of function fftshift.m which puts the negative frequencies before thepositive frequencies. The reason for using it is that when MATLAB function fft.m computesFFT, it outputs the positive frequency components first and then outputs the negativefrequency components.

Below we show full MATLAB code for this exercise and all the figures created startingwith an initial Gaussian pulse. In Figs. B.8–B.10, we show Gaussian pulse in time and itsFourier transform. It was obtained with the following MATLAB program.

% File name: FT_example_2.m

% Testing Fourier transform in Matlab

clear all;

x=-4:0.1:4; % Interval of independent variable

y=exp(-x.^2); % Define Gaussian function

plot(y,’LineWidth’,1.5); % Plot Gaussian (Fig.1)

xlabel(’Number of points’,’FontSize’,14); ylabel(’y(x)’,’FontSize’,14);


445 FFT in MATLAB

0 20 40 60 80 1000

5

10

15

20abs(F)

0 20 40 60 80 1000

5

10

15

20

Spatial frequency

|F|

FT of a Gaussian: fftshift(abs(F)))

Fig. B.9 The next steps in the application of Fourier transform to the Gaussian pulse. After application of function abs() (left),and f f tshi f t() (right).

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1

1.2

1.4ifft(abs(F))

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1

1.2

1.4Inverse FT: fftshift(ifft(abs(F)))

number of points

y(x)

Fig. B.10 Steps in producing inverse Fourier transform of an original Gaussian pulse. After application of i f f t() (left), andafter application of fftshift(ifft()) (right).

title(’y=exp(-x^2)’);

pause

%

F=fft(y); % Calculate FFT

plot(F,’LineWidth’,1.5); % Plot of FT (Fig.2)

title(’F=fft(y)’);

pause

%

plot(abs(F),’LineWidth’,1.5); % Plot of abs value (Fig.3)


title(’abs(F)’); % Take absolute value and plot

pause

%

plot(fftshift(abs(F)),’LineWidth’,1.5); % Use Matlab fftshift function

xlabel( ’Spatial frequency’); ylabel(’|F|’); % (Fig.4)

title(’FT of a Gaussian: fftshift(abs(F)))’);

pause

%--------------------------------------------------------

% Take inverse Fourier transform

%--------------------------------------------------------

plot(ifft(abs(F)),’LineWidth’,1.5); % Use Matlab function ifft (Fig.5)

title(’ifft(abs(F))’);

pause

%

plot(fftshift(ifft(abs(F))),’LineWidth’,1.5); % Use functions fftshift

title(’Inverse FT: fftshift(ifft(abs(F)))’); % and ifft (Fig.6)

xlabel(’number of points’); ylabel(’y(x)’);

pause

close all

B.11 Problems

1. Use Newton’s method to find roots of the following function: ex = 1 − cos3x. Plot itfirst and use the graph to determine the starting point used by Newton’s method.

2. Verify orthogonality relations, Eqs. (B.35) and (B.36) (where the limits of integrationare [0, 2π ]) for another limit, i.e. [−π, π].

3. Derive expressions for coefficients an, bn, cn using new limits.4. Show that the Crank-Nicolson scheme is accurate to O(�t)2.5. Use the 2D version of Euler method to analyse projectile motion [3].6. Analyse the effect of scaling within Fourier transform in MATLAB.

References

[1] S. E. Koonin. Computational Physics. The Benjamin/Cummings, Menlo Park, CA,1986.

[2] P. L. DeVries. A First Course in Computational Physics. John Wiley & Sons, NewYork, 1994.

[3] A. L. Garcia. Numerical Methods for Physics. Second Edition. Prentice Hall, UpperSaddle River, NJ, 2000.

447 References

[4] C. F. Garald and P. O. Wheatley. Applied Numerical Analysis. Addison-Wesley,Reading, MA, 1999.

[5] S. S. Rao. Applied Numerical Methods for Engineers and Scientists. Prentice Hall,Upper Saddle River, NJ, 2002.

[6] M. T. Heath. Scientific Computing. An Introductory Survey. Second Edition. McGraw-Hill, Boston, MA, 2002.

[7] G. W. Recktenwald. Numerical Methods with Matlab. Implementations and Applica-tions. Prentice Hall, Upper Saddle River, NJ, 2000.

[8] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery. Numerical Recipesin Fortran 90. The Art of Scientific Computing. Second Edition. Cambridge UniversityPress, Cambridge, 1992.

[9] L. V. Fausett. Applied Numerical Analysis Using MATLAB. Prentice Hall, UpperSaddle River, NJ, 1999.

[10] C. F. van Loan. Introduction to Scientific Computing. Second Edition. Prentice-Hall,Upper Saddle River, NJ, 2000.

[11] T. L. Harman, J. Dabney, and N. Richert. Advanced Engineering Mathematics withMATLAB. Brooks/Cole, Pacific Grove, 2000.

[12] B. Bradie. A Friendly Introduction to Numerical Analysis. Pearson Prentice Hall,Upper Saddle River, NJ, 2006.

[13] I. Barrodale and K. B. Wilson. A Fortran program for solving a nonlinear equationby Muller’s method. Journal of Computational and Applied Mathematics, 4:159–66,1978.

[14] K. E. Atkinson. An Introduction to Numerical Analysis. Second Edition. Wiley, NewYork, 1989.

[15] G. Arfken. Mathematical Methods for Physicists. Academic Press, Orlando, FL, 1985.

Index

absorptioncoefficient, 181in a two-level system, 168, 169infrared, 109of power in photodetectors, 242spectrum, 243ultraviolet, 109

acceptance angle, see critical angleactive region, 173, 176

in a VCSEL, 173air mass, 369amplifier gain, 206amplifier, erbium-doped fibre (EDFA),

209–13typical characteristics, 213, 215

amplifier, semiconductor (SOA)Fabry-Perot (FPA), 223travelling-wave (TWA), 223, 227

antireflection (AR) coating, 50–1, 205half-wave layer, 53quarter-wave layer, 53

array waveguide grating (AWG), 318asymmetry parameter, 67attenuation, see loss

band gap, 243in solar cells, 376

bandwidth3-dB, 207of a photodetector, 247of an NRZ signal, 144of an optical amplifier, 207of an RZ signal, 143of gain, 206, 224

Bessel functionsmodified, 113, 114, 129, 132, 133ordinary, 113, 129, 131, 132

bit error rate (BER), 240, 252–7, 259,336

bit-pattern independency, 242bit-rate transparency, 241boundary conditions

absorbing (ABC) in 1D, 272–5absorbing (ABC) in 2D, 277–9for a magnetic field, 37–8for an electric field, 36–7Mur’s first order, 272–3

Bragg mirror, 53, 54, 62reflectivity spectrum, 57, 62

Brewster’s angle, 49–50

C-band, see transmission bandscarrier

generation rate, 182leakage rate, 183lifetime, 184recombination rate, 183

chirping, 192–3cloaking, 398–9coefficient of finesse, 29coordinate transformation, 398Courant-Friedrichs-Levy (CFL) stability

condition, 270Crank-Nicolson scheme, 301, 302critical angle, 18, 30, 64, 106cross-gain modulation (XGM), 234–5cross-phase modulation (XPM), 233,

235current

bias, 144dark, 246forward diode, 372injection, 143leakage, 182photocurrent, 242, 245, 371, 374reverse saturation, 374short-circuit, 369threshold, 143

current density, 35cutoff wavelength

in optical fibres, 122–3, 128in photodiodes, 245

Debye medium, 280depletion region, 370detection

coherent, 240incoherent (direct), 240

diffraction, 396dispersion

equation of the waveguide, 66group velocity (GVD), 106, 124in free space, 24material, 124–8, 136, 280

448

449 Index

modal, 124multipath, 108numerical, 269, 270, 283of a 1D wave, 21of a Debye medium, 280of a Lorentz medium, 280of a pulse in optical fibre, 127–8waveguide, 124–127

distributed Bragg reflector (DBR), 173,174

Drude relation, 389dynamic range, 241

E-band, see transmission bandseffective index method, 89–92effective medium, 391, 394effective thickness of a slab waveguide, 77EH modes, 116, 117, 119electric

boundary conditions, 36–37field intensity, 35flux, 35

electron, see carrieremission

amplified spontaneous (ASE), 215, 248,255

spontaneous, 168, 169, 208stimulated, 168, 169

equivalent circuit, 193for bulk laser, 194–6for PIN photodetector, 245, 250for solar cell, 373–6, 380

error function, 231Euler differentiation method, 429–31Euler’s rule (integration method), 436Euler-Cromer differentiation method, 431evanescent waves, 396excited state, 168external modulator, 6, 193eye diagram, 337

Fabry-Perot (FP)interferometer, 29–30resonance conditions, 171resonant modes, 171, 172resonator, 170

Faraday rotator, 323Fast Fourier transform (FFT), 443–4Fermi golden rule, 180Fermi level, 176fibre Bragg grating, 318filter function, 258finite difference method, 425–9

backward differences, 426central differences, 426forward differences, 426

finite-difference (FD) approximation, 297–9

focallength, 19, 20plane, 19, 20point, 19

four-wave mixing (FWM), 233, 352Fourier series, 438–9

change of interval, 439–40of a square wave, 440–1

Fourier transform, 441–2of a chirped Gaussian pulse, 142of a Gaussian pulse, 141of a rectangular pulse, 139, 140, 158of an electric field, 146

Fresnelcoefficients and phases for TE polarization, 44–7coefficients and phases for TM polarization,

47–8reflection, 45, 47

Fresnel approximation, 290

gainapproximate formula, 206differential, 178in an EDFA, 213in semiconductors, 177–81peak, 178, 180ripple, 227, 237saturation, 190, 207, 216, 234spectrum of semiconductor laser, 172

Gaussian probability density function (pdf), 255generation of carriers

in laser didoes, 183in PIN photodiodes, 244in solar cells, 371

geometrical optics, 17–21Goos-Hanchen (G-H) shift, 58–9, 397graded index, 20, 107grid, see meshGRIN system, 20–1group

delay, 124–5, 148velocity, 32

guides modesin optical fibres, 114in slab waveguides, 65–6

HE modes, 116, 117, 119Helmholtz equation, 288, 289, 292, 299hole, see carrierhybrid modes, 116, 119

impedance, 41interference

in dielectric films, 26–7intersymbol (ISI), 255multiple, 27–9

invisibility cloak, see cloaking

450 Index

L-band, see transmission bandslarge-signal analysis (in laser diodes), 192laser diode

distributed feedback laser (DFB), 173, 174in-plane laser, 172vertical cavity surface-emitting laser (VCSEL),

172, 173leap-frog differentiation method, 431left-handed materials (LHM), see negative index

materials (NIM)lens

perfect, 396thin, 19

linear system, 338linewidth enhancement factor (Henry factor), 228Lorentz medium, 280loss

extrinsic, 110in optical fibres, 108, 129intrinsic, 109–10

LP modes, 119–20

Mach-Zehnder interferometer (MZI), 235magic time step, 270magnetic

boundary conditions, 37–8field intensity, 35flux, 35

Maxwell’s equations, 262differential form, 35in cylindrical coordinates, 111–12integral form, 36source-free, 69, 388

mesh1D generation algorithm, 102staggered grid, 266Yee grid in 1D, 266, 267Yee grid in 2D, 276

mid-point differentiation method, 431mode number

azimuthal, 116radial, 116

modulation formatnon-return-to-zero (NRZ), 143–5return-to-zero (RZ), 143–5

modulation of semiconductor lasers, 143, 144modulation response function, 188, 190–2Muller’s method, 100, 422–5

negative index materials (NIM), 385, 386Newton’s method, 420–1noise

Gaussian, 255–7in optical amplifiers, 208–9in photodetectors, 248–9shot, 249thermal (Johnson), 249–50

nonlinear Schrodinger equation, 353–7normalized guide index, 67, 94Numerical aperture (NA), 65, 106–7

O-band, see transmission bandsoptical black holes, 399–400optical cavity, 168

Fabry-Perot, 170optical communication system, 331–3optical coupler, 319–22optical fibre

bit-rate, 108extrinsic loss, 110intrinsic loss, 109–10modes, 116–17single-mode, 122–3

optical isolator, 322–3optical splitter, 319–20

p-n junctiondouble heterostructure, 176–7homogeneous (homojunction), 175–6multijunction, 376–7

paraxial approximation, 288–92permeability, 35

negative, 386, 391–5permittivity, 35

negative, 387, 389–91, 395plasmon-like, 386

photocurrent, 245in photodiodes, 242in solar cells, 370, 371, 374

photodiodeavalanche (APD), 242, 248, 258metal-semiconductor-metal (MSM),

246PIN, 242, 244power, 252sensitivity, 241, 257

photondensity, 183lifetime, 183, 184, 187

Planck’s radiation law, 169plane wave , 40–1plasma, 390

frequency, 390, 391Poisson distribution, 249, 252, 253polarization

circular, 42–3elliptical, 42–3linear, 42transverse electric (TE), 44transverse electric (TM), 47

polarization of dielectric medium, 148population inversion, 167, 204, 208power budget, 333–4Poynting vector, 59–60

451 Index

propagation constant, 41, 42in slab waveguides, 66, 95

pulse broadening, 12, 151in optical fibres, 106

pulse half-width, 142pulse type

chirped Gaussian, 141–2, 159Gaussian, 139–40, 159, 163, 164, 281–6rectangular, 138–9, 157, 340–2, 344Super-Gaussian, 140–1, 159, 162

pulse wave, see waveformpumping, 167, 168

quantum efficiency, 247quantum well, 173, 232quasi-Fermi level, 175, 176

rate equations in EDFA, 211rate equations in laser diodes

for an electric field, 184–7for carriers, 182–3for photons, 183parameters, 184

ray optics, 17in metamaterials, 389in slab waveguides, 64–9

Rayleigh scattering, 110receiver, 331, 343recombination of carriers, 183reflection

at a plane interface, 17, 18coefficient, 17, 45, 46, 48external, 48, 49internal, 48of TE polarized waves, 44–7, 61of TM polarized waves, 48, 61total internal, see critical angle

refractive index, 17in GRIN structures, 20negative, 384numerical values for popular materials, 18relative difference, 65

relaxation-oscillation frequency, 189Resonant cavity, see optical cavityrise time, 144rise time budget, 333–6Runge-Kutta method

fourth order, 433second order, 432

S-band, see transmission bandsSellmeier equation, 31, 125–6, 135signal-to-noise ratio (SNR), 208, 248,

336Simpson’s rule (integration method), 437–8slowly varying envelope approximation (SVEA), 150,

296–7

small-signal analysis (in laser diodes)with linear gain, 188–9with non-linear gain, 189–92

Snell’s law, 18soliton

interactions, 363period, 362

spectralintensity of solar energy, 368, 369responsivity, 247

split-ring resonator (SRR), 391–4split-step Fourier method, 357–60steady-state analysis

in a laser diode, 187in an EDFA, 211–12

step index, 107Stokes relations, 24–6susceptibility, 186

TE modesin optical fibres, 116–18in slab waveguides, 71–2

three-level system, 210threshold

carrier density, 178, 182current, 143

time constant in photodetectors, 246, 247time division multiplexing (TDM), 11time-harmonic field, 39–41TM modes

in optical fibres, 116–18in slab waveguides, 71–2

train of pulses, see waveformtransatlantic telecommunications cable (TAT), 6transfer function, 339transfer matrix approach

for antireflection (AR) coatings, 51–3for Bragg mirrors, 54–7for slab waveguides, 79–85

transitionsin a two-level system, 169–70in semiconductors, 174–5

transmission bands, 10transmittance of Fabry-Perot interferometer, 29, 33transmitter, 331, 342transparency density, 178transparent boundary conditions, 304–6transverse resonance condition, 66–7

normalized form, 67–9trapezoidal rule (integration method), 437two-level system (TLS), 167, 169

U-band, see transmission bands

velocitygroup, 23–4, 124, 126phase, 21–3, 32

Verlet differentiation method, 432

452 Index

wave equation, 38–9for TE modes, 72for TM modes, 72in cylindrical coordinates, 112in metamaterials, 388

waveform, 145, 160waveguide

2D, 88–92asymmetric slab (planar), 75–9, 95cylindrical (optical fibre), 110–23lossy, 86

symmetric slab (planar), 72–5wavevector, see propagation constantweakly guiding approximation (wga),

118–19wire medium, 395

Y-junction, 14, 325, 327Yee algorithm

lossless in 1D, 266–8, 282lossless in 2D, 275–7, 285lossy in 1D, 271–2

Documents

Computational Photonics, An Introduction With MATLAB